Matching Polynomials of Symmetric, Semisymmetric, Double Group Graphs, Polyacenes, Wheels, Fans, and Symmetric Solids in Third and Higher Dimensions

Abstract

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials.

1. Introduction

Matching polynomials and the related perfect matchings of structures, lattices, and graphs have intrigued scientists and mathematicians over decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. These polynomials have wide-ranging applications, spanning aromaticity [20,21,22,23,24,25,26,27] and Ising models [28,29,30,31,32]. These polynomials find important applications in the combinatorial enumeration of resonance structures and related graph theory pertinent to aromaticity and superaromaticity [33,34,35,36,37,38,39,40,41,42,43,44,45,46], the energetic stabilities of π-electronic structures, fullerene cages, nanospheres, and nanomaterials [47,48,49,50,51,52,53,54,55,56]. For example, fullerene cages [47,48,49,50,51,52,53,54,55,56], which comprise 12 pentagons and a varied number of hexagons, have revived a growing interest in matching polynomials and the concept of aromaticity. The stabilities and π-electronic structural features of these cages, nanospheres, and golden fullerenes can be more readily understood through generalized definitions of aromaticity that encompass these structures and combinatorial enumeration techniques. The celebrated buckminsterfullerene (C₆₀) exhibits 1812 isomers and its sister C₇₀ fullerene exhibits a combinatorics of 8149 isomers [47,48,49,50,51,52,53,54,55,56]. Their electronic, magnetic, and spectral properties can be interpreted using combinatorial methods. Furthermore, shape characterization and related topics often need such mathematical tools for the computation and prediction of properties of chemical structures and cages [57,58]. Recent years have witnessed the blossoming of machine learning and artificial intelligence techniques, opening the gateway to a plethora of such applications, many of which require graph theoretical and combinatorial tools [59].

Matchings in graphs and structures are simply the number of ways to place k disjoint dimers or coloring k disjoint edges in the graph, such that no two vertices in any of the colored edges are neighbors. In a sense, this is a combinatorial problem of match-making from an array of networked persons such that the couples (dimers) are in a committed/monogamous relationship. In the chemical context, this could correspond to the dimer chemisorptions with nearest neighbor exclusions or the Kekulé resonance structures of polycyclic aromatic compounds and so forth. In graph theory, matchings provide important insights into the connectivity, combinatorial complexity, and nature of the graphs and networks. Consequently, the topic of matching polynomials has been pioneered by Harary, Farrell, Hosoya, and many others [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Recently, the present author [1,2] has studied graph polynomials with a renewed definition of aromaticity through delta polynomials and exploration into the connections with several orthogonal polynomials [60].

A number of computational and mathematical techniques have been considered through the use of combinatorics and graph theory related to these polynomials. For example, the enumeration of conjugated circuits, the enumeration of Kekulé structures, and sextet polynomials are all related to the problem of matching polynomials [43,46]. Carbon nanospheres and giant golden fullerenes have rekindled further interest in such combinatorial methods because high-level ab initio quantum chemical studies on such systems can be intensive and challenging. Hence, there is a clear and compelling need for combinatorial and graph theoretical techniques for such large nanomaterials in order to shed light on their properties, structures, stabilities, and various spectra. It has been shown that there appears to be a very strong correlation between the various combinatorially related entities, such as topological resonance energies, conjugated circuits, and sextet polynomials, and the relative stabilities of these structures. Furthermore, graph theory can be applied to the enumeration of isomerization or rearrangement pathways, for example, the Stone–Wales rearrangement graphs [51] and other isomerization graphs of nonrigid isomers of polysubstituted organic compounds, water clusters, and other fluxional molecules exhibiting pseudorotations [61,62,63,64,65,66,67].

The objective of the present study is the computation of matching polynomials of symmetric, semisymmetric, and double group graphs that are particularly intriguing or have been the subject matter of several studies over the decades, especially in the context of graph theory. The computations of matching polynomials of such graphs with clustered or fused cycles are computationally intensive beyond the NP problem because of the combinatorial complexity of the graphs. The number of steps in recursive algorithms and hence the number of computations grow astronomically for graphs containing multiple fused cycles and clustered graphs in n-dimensions. For such complex graphs, there are no polynomial algorithms for the computation of the matching polynomials. Each step of recursion is akin to the bacterial cell division process, and this underscores the complexity of computing the matching polynomials, which do not scale in any polynomial algorithmic steps. Consequently, most of the previous studies are limited to graphs containing fewer vertices, especially those that are not clustered with multiple fused cycles. The present study explicitly considers graphs and solids in higher dimensions that are extremely clustered, thus posing much greater computational challenges for obtaining the matching polynomials. Moreover, we have shown that the results obtained earlier in the literature [20] for the matching polynomials of some of these graphs are erroneous, and we provide new corrected results. We have also considered the matching polynomials of some challenging graphs such as Mobius graphs, semisymmetric graphs, and a variety of solids in three and higher dimensions. The Möbius graphs have also inspired the novel stereochemical synthesis of twisted molecules in the general context of Möbius combinatorial chemistry [67,68,69]. The double group graphs can be of great interest in the field of relativistic quantum chemistry, where the introduction of spin–orbit coupling results in a double group of molecular point group symmetries [70,71,72]. Such double groups have also been considered in the context of quarks [73]. The present study is generalized to encompass structures in four and five dimensions [74,75], as well as polyacene series, fans, and wheels, all of which constitute interesting recursive structures. Some of the graphs considered here have religious symbolisms [76,77] and their tessellations have also been considered in material science [78]. Matching polynomials of several such structures are computed in this study for the first time. Moreover, some interesting connections to a variety of combinatorial numbers and orthogonal polynomials are pointed out. With the exception of a few simple graphs, the matching polynomials of several complex graphs are computed and reported here for the first time. The novelty of the current study lies in the complex computations of several newly reported matching polynomials and several combinatorial connections that are revealed as orthogonal polynomials and combinatorial sequences.

2. Mathematical and Computational Methods Pertinent to Matching Polynomials

The matching polynomial, which is also sometimes referred to as the acyclic polynomial of a graph G, is defined as

$$M_G\left(x\right)={\sum }_{k^{\prime }=0}^{\left[{\displaystyle \frac{n}{2}}\right]}{\left(-1\right)}^{k^{\prime }}p\left(G,k^{\prime }\right)x^{n-2k^{\prime }}={\sum }_{k=0;even}^{\left[{\displaystyle \frac{n}{2}}\right]}{\left(-1\right)}^k{C}_k{x}^{n-k},$$

(1)

where p(G, k’) enumerates the number of ways to place k’ disjoint dimers on the graph, and n is the number of vertices of the graph while [n/2] is the greatest integer contained in n/2. We introduce k = 2k’ for convenience, which implies k must always be even, and the sum over k is restricted to even k in the above expression. Hence, all matching polynomials are displayed using the last part of the above expression with k = 0, 2, 4, …[n/2]. Among several chemical structures, fullerenes are chemically interesting carbon cages with exactly 12 pentagons and a varied number of hexagons, and it readily follows that n is even for fullerenes. This also applies to several chemically interesting polyhedra and D_2h-polyacenes. It can be shown that the characteristic and matching polynomials are identical for trees; hence, the synonym acyclic polynomials applies to the matching polynomials. An interesting byproduct of matching polynomials is the constant coefficient or C_[n/2], which enumerates the number of perfect matchings. Therefore, the constant coefficient in the matching polynomial also yields the number of Kekulé structures for polycyclic aromatic compounds containing fused benzene rings.

Matching polynomials are invariant to the labeling of the vertices of the graph and in this sense, these polynomials offer yet another independent structural invariant. The matching polynomials form the basis for the Z-index, a topological invariant formulated by Hosoya [9]. The Z-index and matching polynomials complement other structural invariants, usually called the topological indices. Such indices are derived either from the distance matrices or vertex degrees of graphs. The matching polynomials are quite different in that they enumerate the number of k-matchings in the graph, and hence they are not directly related to the distance matrices of the graphs. There exists a relation between the perffians, which are symmetric versions of the pfaffians of adjacency matrices of the graphs, and the number of perfect matchings. This number corresponds to the constant coefficient of the matching polynomial. It should be noted that the computation of perffians is an NP problem, as it requires evaluating permanents involving n! permutations. Consequently, even the computation of the coefficient of the constant term of the matching polynomial is an NP problem. This in turn reveals the computational intensity and complexity of computing all coefficients of the matching polynomials.

The computations of the matching polynomials of simple graphs such as paths, trees, monocyclic graphs, Cayley trees, Bethe lattices, or graphs containing simple cycles connected by edges or spirographs are somewhat less complex. Hence, their matching polynomials have been computed in the literature [10,11]. The required computations are far more complex for clustered graphs containing multiple fused rings, twisted or nonplanar graphs, three-dimensional clusters, and clustered graphs, as well as in higher dimensions such as nd-hypercubes [74,75], nd-polyhedra, isomerization graphs, cages, fused polycyclic nanomaterials, fullerenes, and cage structures. For such highly clustered graphs, the computation of matching polynomials is both CPU- and disk-intensive because there are no polynomial algorithms for such clustered graphs in general. Several algorithms have been considered for the computation of matching polynomials and perfect matchings over several decades [4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Most of these techniques involve some type of recursive reduction or pruning technique of the parent structure through multiple iterative steps so that the pruned fragments are tree graphs or spirographs, where spirographs are graphs with monocycles connected by edges. Hence, there are no fused rings in spirographs. Once such pruned structures are achieved, computational techniques can iteratively assemble the matching polynomials of the parent structure that we started with. We note during our computational procedures that the labeling of the parent graph, or alternatively the order in which the edges of the graph are removed during the recursive reduction, strongly influences the CPU intensity and the storage required for the computations of matching polynomials of combinatorially complex structures. In the present study, we employ a series of algorithms that can handle complex clustered graphs such as fullerenes and symmetric, semisymmetric, and isomerization graphs and cages.

We also demonstrate the utility of the computational techniques by computing the matching polynomials of graphs in the fourth and fifth dimensions, as well as polycyclics containing a large number of vertices. This was accomplished through the synthesis of optimal vertex labeling and optimized recursive reductions, along with the prior computation and storage of matching polynomials of recurring fragments and trees so that they do not have to be computed repeatedly. In this sense, we call these direct computational methods compared to simple binomial recursions that result in a power series of 2^k fragments, where k grows in astronomical proportions for highly clustered graphs and multi-dimensional structures. We have employed quadruple precision arithmetic, which ensures an accuracy of up to 33–35 digits for the coefficients in the matching polynomials. We have exemplified the power of computational techniques with a variety of intriguing simple and complex graphs, such as the Petersen graph, Desargues–Levi graph, Grotzsch graph, Chàvatal graph, Möbius–Kantor graph, Star of David graph, Folkman graph, Coxeter graph, Tutte–Coxeter graph, bislit cube, bidiakis cube, icosahedron, octahedron, 3-cube, 4-cube, 4-tetrahedron, 4-octahedron, 4-icosahedron, 5-cube, 5-tetrahedron, rhombicuboctahedron, dodecahedron, truncated dodecahedron, prisms, D_2h-polyacene series, wheels, fans, and so forth. These graphs were chosen for the present study because they present interesting cases of symmetry, semisymmetry, and double group symmetry and they have interesting applications in isomerization pathways, dynamic chirality, water clusters, aromaticity, chemisorption, and many other associated chemical phenomena.

3. Results and Discussion

A.

Matching Polynomials of the Petersen graph, Chàvatal graph, Grotzsch graph, Star of David graph, Möbius–Kantor graph, Folkman graph, Desargues–Levi graph, Coxeter graph, and Tutte–Coxeter graph

We start with both mathematically and chemically interesting graphs in this section, which are listed in Figure 1. Several graphs in other figures, as well as the graphs in Figure 1, were reproduced from refs [79,80,81,82,83,84,85,86,87]. For each graph, we provide some background information highlighting the applications of the graphs. While a few of these graphs have been considered for matching polynomials, either the results were erroneous [20] or they are included in the present study for systematic and comprehensive comparisons. Some of the graphs shown in Figure 1 are nonplanar and some have been of considerable interest from the viewpoints of stereochemistry, the dynamics of fluxional molecules, isomerization reaction graphs, chemically interesting twisted topologies, and so forth.

Table 1. Matching Polynomials of the Petersen graph, Chàvatal graph, Grotzsch graph, Star of David graph, Desargues–Levi graph, Folkman graph, Coxeter graph, and Tutte–Coxeter Graph.

Petersen Graph
Petersen Graph S5 (120)
Xn−k	Ck
0	1
2	−15
4	75
6	−145
8	90
10	−6
Matching Polynomial of the Chàvatal Graph (D4)
Xn−k	Ck
0	1
2	−24
4	204
6	−752
8	1175
10	−628
12	52
Matching Polynomial of the Grotzsch Graph
Xn−k	Ck
0	1
2	−20
4	135
6	−365
8	360
10	−87
Matching Polynomial of the Star of David Graph
Xn−k	Ck
0	1
2	−18
4	111
6	−276
8	255
10	−66
12	2
Matching Polynomial of the Möbius–Kantor Graph: Oh2 (Order:96)-Double Group of the Octahedral Group
Xn−k	Ck
0	1
2	−24
4	228
6	−1096
8	2826
10	−3816
12	2444
14	−600
16	33
Matching Polynomial of the Semisymmetric Folkman Graph: S5[S2]: O(3840)
Xn−k	Ck
0	1
2	−40
4	660
6	−5840
8	30,200
10	−93,408
12	170,080
14	−172,480
16	86,880
18	−17,280
20	768
The Desargues–Levi Graph (S5 × S2)
Xn−k	Ck
0	1
2	−30 (2 × 3 × 5)
4	375 (3 × 53)
6	−2540 (22 × 5 × 127)
8	10,155 (3 × 5 × 677)
10	−24,486 (2 × 3 × 7 × 11 × 53)
12	34,945 (2 × 29 × 241)
14	−27,840
16	11,040
18	−1720 (23 × 5 × 43)
20	60 (22 × 3 × 5)
The Levi-30 Graph:
Xn−k	Ck
0	1
2	−45
4	890
6	−10,215
8	75,665
10	−380,559
12	1,331,660
14	−3,268,585
16	5,604,695
18	−6,610,695
20	5,213,550
22	−2,626,805
24	786,360
26	−123,630
28	8040
30	−120
Matching Polynomial of the Coxeter Graph: PGL2(7) O:338
Xn−k	Ck
0	1
2	−42
4	777
6	−8344
8	57,708
10	−269,640
12	868,700
14	1,934,712 (23 × 33 × 132 × 53)
16	2,942,247
18	−2,970,310
20	1,894,851
22	−703,080
24	130,872
26	−8904
28	84
Matching Polynomial of the Tutte–Coxeter Graph S6 × S2: O(1440)
Xn−k	Ck
0	1
2	−45
4	900
6	−10,560
8	80,820
10	−424,404
12	1,566,360
14	−4,094,280
16	7,541,460
18	−9,622,660
20	8,246,160
22	−4,517,280
24	1,464,120
26	−247,320
28	17,280
30	−288 (2 × 122)= ${\displaystyle \frac{1}{2}}{\left[\begin{array}{c}5\\ 1\end{array}\right]}^2$

shows the matching polynomials of all graphs in Figure 1 in the same order as they appear in Figure 1. We start with the celebrated Petersen graph, which appears in the pseudorotation of trigonal bipyramid complexes such as PF₅. The pseudorotation follows Berry’s mechanism, resulting in an isomerization graph that is isomorphic with the Petersen graph without chirality and its double, namely the Desargues–Levi graph, when chirality is included [62,63,64,65]. Hence, the graph is of particular interest in the context of dynamical stereochemistry, as well as in the field of graph theory. This graph has 10 vertices and 15 edges and gives rise to six perfect matchings, as seen in Table 1. The number of perfect matchings for all graphs considered here is given by the constant coefficient in the matching polynomial or k = n when n is even. Of course, there is no perfect matching when n is odd, as in the case of the Grotzsch graph, which we discuss subsequently. The Petersen graph has been considered earlier in multiple works [17,62,63,64,65]; thus, the polynomial for the Petersen graph in Table 1 for this graph simply validates the well-known results.

Next, we consider the Chàvatal graph, which contains 12 vertices and 24 edges. It has been considered a very interesting example of the Erdös conjecture concerning the connection between the chromatic number and k-regularity of certain graphs. The Chàvatal graph is a demonstration of the four-color problem in that it can be colored with four colors and not three and it is a 4-regular, high-girth graph with a girth of 4. The graph exhibits an automorphism group isomorphic with the D₄ symmetry containing eight symmetry operations. It is a regular, Hamiltonian, and Eulerian graph with no triangles. All vertices have the same distance degree sequence of {1,4,7}. Thus, it presents an interesting connectivity case for the matching polynomial. The result in Table 1 suggests that there are 52 perfect matchings for the Chàvatal graph, which is a large number for a graph containing only 12 vertices with the exception of K₁₂ and a few such highly clustered graphs. The peak coefficient for the matching polynomial is reached at k = 8 and the number of 8-dimer covers for the Chàvatal graph is 1175. Unfortunately, many of the previous results for the matching polynomials of this graph and several other graphs are erroneous [20]. This could be primarily because the authors of ref [20] do not use any reliable algorithms and the results are merely deduced through trial observations and deductions.

The Grotzsch graph (Figure 1) contains 11 vertices and 20 edges and possesses a chromatic number of 4. Hence, it is the smallest graph with a chromatic number of 4. It is one of the rare graphs that we consider here with an odd number of vertices; hence, although it does not contain a perfect matching, there are 87 matchings that contain five dimers, leaving out the odd vertex, as inferred from the coefficient of x in Table 1 for the Grotzsch graph. Once again, some of the previously reported coefficients for the matching polynomial of the Grotzsch graph [20] are erroneous. In general, the higher the vertex connectivity, the greater the possibility of placing disjoint dimers on the graph or larger coefficients in the matching polynomial of strongly connected graphs.

The Star of David graph appears in a number of applications, and it can be tessellated further into a network containing multiple units of the graph. The graph holds religious symbolism both in Judaism and Hinduism, where in the former, it is interpreted by the name of the graph. In Hinduism, it represents the tantric energy union of Shiva and Shakthi and it is called the shatkona yantra. The yantra or the instrument is central to the geometric patterns called kolams that are displayed at the entry thresholds of homes in India, especially in the less auspicious month of the second half of December through to the first half of January or Margazhi [76,77]. The degree-based topological indices have been obtained for the networks arising from the Star of David graph and hexagonal cage networks obtained from the graph [78]. This graph contains 12 vertices and 18 edges comprising several triangles. The matching polynomial of this graph is interesting in that it reveals that there are only two perfect matchings for the graph analogous to the number of perfect matchings of C_n, a graph of a monocycle of n even vertices with C₆ representing benzene; the two Kekulé structures of benzene are the two perfect matching analogs of the Star of David graph. Hence, the Star of David graph appears to simulate the union of the two interpenetrating structures into one, reminiscent of a monocycle’s perfect matchings.

The Möbius–Kantor graph presents a very interesting example as it relates to the stereochemistry of twisted molecules, Möbius ladders, and so forth. The embeddings of the Möbius–Kantor graphs, such as the one shown in Figure 1, have been considered [67] as interesting candidates for the potential synthesis of twisted molecules with interesting stereochemistry [68,69]. The graph is one of the smallest graphs exhibiting a twisted structure. It contains 16 vertices and 24 edges. The twisted nature of the graph results in an automorphism group that is isomorphic with the double group of the octahedral group or O_h² containing 96 symmetry operations, making it both an interesting and unusual candidate for the study of matching polynomials. Furthermore, the introduction of spin–orbit coupling in relativistic quantum chemistry requires the treatment of molecular symmetries in the double group. For this reason, the spin representations have been considered in the double group for both icosahedral and nonrigid molecules [70,71]. Thus, double group automorphisms are relativistic molecular spinor symmetry representations. Furthermore, the Cayley graph of the Pauli group is the Möbius–Kantor graph, accentuating the role of Möbius graphs in spinor representations and relativistic quantum mechanics. Moreover, the Möbius–Kantor graph is a subgraph of the 4D hypercube with 384 automorphisms comprising the wrath product S₄[S₂]. For these reasons, we have taken up this graph for the study of matching polynomials. As seen from Table 1, the graph contains an odd number of perfect matchings or 33 perfect matchings with a peak value of 3816 for the five-dimer covers.

Next, we consider the Folkman graph with 20 vertices and 40 edges, genus 3, which is also 4-regular. Although the edges of the Folkman graph all form one equivalence class under the automorphism group of the graph, the vertices are not all equivalent. Hence, this graph is an exemplary illustration of a semisymmetric graph, comprising two equivalence classes of vertex partitions even though all vertices have the same degree of 4. The automorphism group of the Folkman graph is the wreath product S₅[S₂] with 3840 vertex permutations. The 20 vertices of the Folkman graph are partitioned into 10² partitions, rendering it the smallest semisymmetric graph. The distance degree sequences of the Folkman graph are 10 {1, 4, 6, 6, 3} and 10 {1, 4, 9, 6}, consistent with the 10² automorphic partition of the vertices. The graph is bipartite; hence, it exhibits spectral mirror symmetry in accord with the Coulson–Rushbrooke theorem. As seen from Table 1, the Folkman graph exhibits 768 perfect matchings with a peak number of 172,480 matchings for the seven-dimer covers. Although the Folkman graph exhibits an automorphism group isomorphic to the 5D hypercube, it has a smaller number of matchings compared to the 5-cube primarily due to its smaller size and a less clustered nature.

The Desargues–Levi graph, which is the double of the Petersen graph, has been the cynosure of dynamic stereochemistry pertinent to nucleophilic displacement reactions and pseudorotations in trigonal bipyramidal complexes following the pioneering work of Mislow and coworkers [62,63]. The isomerization pathways of certain carbonium ions and phosphorus compounds, which become fluxional due to low potential energy barriers, were shown to be represented by the Desargues–Levi graph. The Petersen graph and the Desargues–Levi graph are two relatives that have found the most extensive applications in fluxional molecules. The Desargues–Levi graph is a cubic graph with 20 vertices and 30 edges. It is simply the generalized Petersen graph G(10,3). It is a symmetric graph with the automorphism group S₅ × S₂ and consequently, it is called the double of the Petersen graph, which is characterized by the automorphism group S₅. It exhibits an integral mirror symmetric spectral pattern; the mirror symmetry arises from its bipartite nature. We have computed the matching polynomial of the Desargues–Levi graph, which is shown in Table 1. As seen from Table 1, the number of perfect matchings of the Desargues–Levi graph is 120, while the peak coefficient in the matching polynomial of 6,610,695 occurs for the nine-dimer covers. The corresponding number of perfect matchings of the Petersen graph is just six, with a peak coefficient of 145 in the case of three-dimer covers.

The last two graphs that we consider in this subsection are the Coxeter graph and the Tutte–Coxeter graph. The Coxeter graph is a symmetric graph with 28 vertices, 42 edges, and a vertex regularity of 3. The automorphism group of this graph is PGL₂(7) with 338 symmetry operations. The graph has a chromatic number of 3 and it is also 3-regular or cubic. On the other hand, the Tutte–Coxeter graph is a 3-regular graph with 30 vertices, 45 edges, a chromatic number of 2, and a girth of 8. The graph exhibits 1440 automorphic symmetry operations, constituting a symmetry group that is isomorphic with the direct product S₆ × S₂. The matching polynomials of both graphs are shown in Table 1. As seen from Table 1, the Coxeter graph exhibits 84 perfect matchings while the Tutte–Coxeter graph exhibits 288 perfect matchings, which can be expressed in terms of the Stirling numbers of the first kind (see Table 1). The peak coefficient of the Coxeter matching polynomial occurs for nine-dimer covers with 2,970,310 matchings. The other interesting factorings of the coefficients are shown in Table 1. The Tutte–Coxeter matching polynomial contains a peak matching coefficient for nine-dimer covers with 9,622,660 matchings.

B.

Matching Polynomials of three-dimensional polyhedra

In Figure 2, we show a few common polyhedra, which we call 3D solids. They present interesting cases for the study of matching polynomials due to their clustered nature. The matching polynomials of some of the structures in Figure 2 have been obtained before by Hosoya [4,6,7,10], the present author [1,2,7,11], and so forth. However, the matching polynomials of all 3D structures considered in ref [20] are erroneous. Therefore, we have included several polyhedral and related structures for the consideration of matching polynomials.

Table 2. Matching Polynomials of a variety of 3D structures: Cube, Octahedron, Icosahedron, Dodecahedron, Bidiakis cube, Bislit cube, Rhombicuboctahedron, and truncated dodecahedron.

Matching Polynomial of Cube
Xn−k	Ck
0	1
2	−12
4	42
6	−44
8	9
Matching Polynomial of Octahedron
Xn−k	Ck
0	1
2	−12
4	30
6	−8
Matching Polynomial of Icosahedron
Xn−k	Ck
0	1
2	−30
4	315
6	−1400
8	2535
10	−1482
12	125
Matching Polynomial of Dodecahedron C20 Ih (120)
Xn−k	Ck
0	1
2	−30
4	375
6	−2540
8	10,155
10	−24,474
12	34,805
14	−27,300
16	10,260
18	−1400
20	36
Matching Polynomial of Bidiakis cube: D4 (8)
Xn−k	Ck
0	1
2	−18
4	117
6	−336
8	418
10	−184
12	12
Matching Polynomial of Bislit Cube (Td: O(24))
Xn−k	Ck
0	1
2	−14
4	55
6	−58
8	9
Matching Polynomial of Rhombicuboctahedron Oh (O:48)
Xn−k	Ck
0	1
2	−48
4	984 (23 × 3 × 41)
6	−11,288
8	79,806
10	−361,248
12	1,054,328
14	−1,951,272
16	2,196,753
18	−1,394,608
20	436,608
22	−51,552 (25 × 32 × 179)
24	1088 (17 × 43)
Matching Polynomial of truncated dodecahedron: Ih (O:120)
Xn−k	Ck
0	1
2	−90
4	3825
6	−102,100
8	1,920,480
10	−27,073,548
12	297,017,670
14	−2,599,271,940
16	18,452,804,370
18	−107,509,368,860
20	518,092,164,744
22	−2,075,424,449,400
24	6,929,555,927,025
26	−19,297,656,051,090
28	44,774,805,188,205
30	−86,315,702,921,360
32	137,639,652,148,260
34	−180,432,784,692,900
36	192,895,567,767,700
38	−166,490,504,865,960
40	114,582,353,107,800
42	−61,926,709,855,920
44	25,792,171,457,280
46	−8,085,072,744,000
48	1,850,294,700,320
50	−296,798,234,112
52	31,509,790,080
54	−2,030,914,560
56	68,820,480
58	−921,600
60	2048

shows our computed matching polynomials.

Our results in Table 2 for the matching polynomials of some of the structures shown in Figure 2 completely agree with those obtained by Hosoya [4] but disagree with the results in ref [20]. Evidently, simple inspections and trial-and-error methods such as the ones employed in ref [20] cannot be relied upon. Although the matching polynomials of the cube and octahedron have been studied before, the icosahedron certainly deserves some merit for consideration. As seen from Table 2, the 12-vertex icosahedron contains 36 perfect matchings. The peak coefficient for the icosahedron occurs for four-dimer covers with 2535 matchings. The icosahedra occur in a number of metal clusters with gold atoms such as Au₁₃ and in borane chemistry. The previously obtained matching polynomials of the bislit cube and bidiakis cube are erroneous, and Table 2 shows the correct results obtained from our computer codes. The matching polynomials of the truncated dodecahedron and rhombicuboctahedron grow in large proportions due to several fused cycles and the clustered nature of these 3D structures. In particular, the numbers of perfect matchings for the two structures are 1088 and 2048. The peak coefficients for the two matching polynomials are 2,196,753 and 192,895,567,767,700, respectively (see Table 2).

C.

Matching Polynomials of three-dimensional, 4D, and 5D structures

In this section, we consider the matching polynomials of the 4D and 5D structures, as they pose a greater challenge compared to the simpler 2D and 3D structures. Moreover, the 4D and 5D structures are more clustered and appear in several chemical applications. Such 4D and 5D structures are displayed in Figure 3. In general, we can construct polyhedral structures in higher dimensions from the corresponding 3D polyhedra using the Cartesian product of two graphs. Let Q_n be the three-dimensional polyhedral structure with the adjacency matrix A_Qn. Then, the corresponding polyhedral graph in the next higher dimension is defined by the graph’s Cartesian product:

Q_n+1 = Q_n × K₂, n > 2,

where K₂ is simply a path of length 2. The adjacency matrix of Q_n+1 would thus be twice the order of Q_n, and the process can be iterated to higher dimensions. This recursive construction can be iterated to any order and thus we arrive at nd-polyhedra in the fourth, fifth, and higher dimensions. The adjacency matrix of Q_n+1, A_Qn+1 is given by

$$A_{Q_{n+1}}=\left[\begin{array}{cc}{A}_{Q_n}& I\\ I& A_{Q_n}\end{array}\right],$$

where I is simply the identity matrix of the same order as A_Qn.

The 4D tetrahedral, 4D octahedra, and 4D icosahedra thus obtained are shown in Figure 3. Another way to visualize these structures is to inscribe a copy of the structure within the structure and join the vertices that correspond to each other (Figure 3).

The matching polynomials obtained after recursive construction of the structures in the fourth and fifth dimensions are shown in

Table 3. Matching polynomials of four-dimensional and a few five-dimensional solids.

Matching polynomial of 4-tetrahedron
Xn−k	Ck
0	1
2	−16
4	72
6	−88
8	16
Matching Polynomial of 4-octahedron
Xn−k	Ck
0	1
2	−30
4	315
6	−1404
8	2571
10	−1518
12	137
Matching Polynomial of 4-cube
Xn−k	Ck
0	1
2	−32
4	400
6	−2496
8	8256
10	−14,208
12	11,648
14	−3712
16	272 (17 × 24)
Matching Polynomial of 4-icosahedron
Xn−k	Ck
0	1
2	−72
4	2196
6	−37,160
8	384,060
10	−2,517,456
12	10,535,152
14	−27,675,936
16	43,782,600
18	−38,663,520
20	16,625,760
22	−2,632,320
24	66,400
Matching Polynomial of 5-tetrahedron
Xn−k	Ck
0	1
2	−40
4	620
6	−4744
8	18,934
10	−38,360
12	35,564
14	−11,768
16	673
Matching Polynomial of 5-cube a
Xn−k	Ck
0	1
2	−80
4	2840
6	−59,120
8	803,580
10	−7,517,264
12	49,715,240
14	−235,146,480
16	795,862,790
18	−1,910,146,160
20	3,190,117,800
22	−3,594,554,960
24	2,605,908,220
26	−1,129,177,840
28	259,084,440
30	−25,108,944
32	589,185 32 × 5 × 13,093

^a perfect matching of n-cube forms a sequence; {3², 17 × 4², 3² × 5 × 13,093…}.

. As seen from Table 3, the number of perfect matchings of a 4-tetrahedron is sixteen compared to three for a regular tetrahedron. Likewise, the number of perfect matchings for a 4-octahedron is one hundred and thirty-seven compared to eight for a regular octahedron. This is understandable owing to the considerably increased complexity of the structures in the fourth dimension compared to the corresponding regular polyhedra. This is the first study that considers the matching polynomial of a 4-icosahedron (Table 3). The number of perfect matchings of a 4-icosahedron is 66,400 compared to 125 for a regular icosahedron, indicating the possibility of an astronomical expansion for the number of matchings as a function of the dimension.

The number of perfect matchings for the n-cube forms the interesting sequence {3², 17 × 4², 3² × 5 × 13,093, 2¹⁵ × 3² × 7 × 7,911,539, …}. The other coefficients of the matching polynomials of the 4-cube and 5-cube in Table 3 increase rapidly with the peak coefficients of the polynomials, forming the sequence {44, 14,208, 3,594,554,960, 1,378,948,882,982,541,631,488, ….}. Moreover, the third coefficient of the matching polynomials of the n-cubes forms the sequence

{2, 42, 400, 2840, 17,376….} = {(1)2, (7)6, (25)16, (71)40, (181)96,…}

for the 2-cube, 3-cube, 4-cube, 5-cube, and 6-cube, respectively. In the second sequence above, the factored-out integer is exactly n × 2ⁿ⁻² for the n-cube. The remaining numbers form the sequence {1, 7, 25, 71, 181,….}. Hence, the first three coefficients for any n-cube are obtainable by using the above sequence.

D.

Matching Polynomials of Prism Graphs and circulenes

Next, we consider a class of chemically interesting graphs called the prism graphs; representatives are shown in Figure 4. Analogous to n-cubes, the prism graphs of any cross-section can be likewise constructed by the Cartesian product

Pr_2n = P₂ × C_n,

where P₂ is a path of length 2 and C_n is the monocyclic graph on n vertices. The prism graph contains 2n vertices and 3n edges. The prism graph is also a precursor to building simple cyclic nanotubes. Thus, they are of interest in characterizing their aromaticity and so forth.

Table 4. Matching Polynomials of prisms and Circulenes: hexagonal prism (6-prism), 20-prism, and [7]circulene.

Matching Polynomial of 6-prism
Xn−k	Ck
0	1
2	−18
4	117
6	−336
8	420
10	−192
12	20
Matching Polynomial of 20-prism
Xn−k	Ck
0	1
2	−60
4	1650
6	−27,580
8	313,355
10	−2,564,152
12	15,624,200
14	−72,305,160
16	256,948,925
18	−704,514,900
20	1,489,893,222
22	−2,418,300,660
24	2,984,254,025
26	−2,759,465,880
28	1,873,096,780
30	−907,342,584
32	301,700,575
34	−65,197,340
36	8,444,350
38	−573,100
40	15,129
Matching Polynomial of [7]Circulene
Xn−k	Ck
0	1
2	−35
4	539
6	−4816
8	27,720
10	−107,912
12	290,339
14	−542,329
16	696,906
18	−602,070
20	335,671
22	−112,728
24	20,223
26	−1540
28	29

shows the computed matching polynomials for large prisms, including the 20-prism. The hexagonal prism has 20 perfect matchings and a peak coefficient of 420 for a four-dimer cover on the hexagonal prism. On the other hand, the 20-prism contains 15,129 perfect matchings and a peak coefficient of 2,984,254,025 for the case of the 12-dimer cover. Our results for the case of the 20-prism clearly demonstrate the power with which computations can be made on such relatively large clustered systems, even though no direct polynomial order algorithm exists for the computation of matching polynomials, unlike the characteristic polynomials which can be more readily computed to the polynomial order of operations.

[7]circulene has received considerable attention from the experimental chemical community as its synthesis has posed some challenges [89]. The molecule is also interesting from aromaticity and energetic stability standpoints, as it is composed of a central heptagonal ring. For this reason, we have computed the matching polynomial of [7]circulene, which is shown in Table 4. The structure has 29 perfect matchings and a peak coefficient of 696,906 for the eight-dimer cover. The results can be compared to the corresponding numbers for coronene, which contains a hexagon in the center as opposed to the heptagon of [7]circulene. The coronene molecule’s matching polynomial reveals that it has 20 perfect matchings and a peak coefficient of 90,792 for a seven-dimer cover. Consequently, although the central ring is composed of a heptagon, the matching polynomial of [7]circulene explains its stability and thus its experimental synthesis and observation.

E.

Matching Polynomials of D_2h Symmetric Polyacenes

The D_2h polyacenes are formed by fusing hexagons in a linear fashion, as shown in Figure 5 with the illustration of dodecacene or a D_2h linear system of 12 fused aromatic rings. The properties of such polyacenes, including their electronic, spectroscopic, and thermodynamic properties as a function of the number of rings, have been of considerable interest over the years. The series starts with benzene, naphthalene, anthracene, tetracene, and so forth. The matching polynomials of such D_2h polyacenes for a large number of hexagons are shown in

Table 5. Matching Polynomials of D_2h Symmetric Polyacene series: C_4n+2H_2n+4.

Matching Polynomial of Dodecacene (n = 12) C50H28
Xn−k	Ck
0	1
2	−61
4	1736
6	−30,626
8	375,366
10	−3,395,178
12	23,498,917
14	−127,361,785
16	548,705,684
18	−1,896,924,530
20	5,290,433,560
22	−11,927,625,748
24	21,721,468,366
26	−31,840,941,462
28	37,338,708,833
30	−34,709,568,941
32	25,258,851,533
34	−14,150,729,127
36	5,968,641,194
38	−1,839,602,672
40	397,649,627
42	−56,858,935
44	4,923,425
46	−223,119
48	3991
50	−13
Matching Polynomial of Icosadecaacene: C82H44
Xn−k	Ck
0	1
2	−101
4	4892
6	−151,318
8	3,358,252
10	−56,961,072
12	768,023,033
14	−8,453,932,129
16	77,422,745,297
18	−598,247,474,531
20	3,941,670,102,182
22	−22,323,819,077,892
24	109,352,563,856,004
26	−465,483,428,654,996
28	1,727,898,071,519,730
30	−5,607,400,985,161,978
32	15,935,046,764,398,403
34	−39,690,091,049,676,019
36	86,663,388,278,077,032
38	−165,813,129,496,595,054
40	277,691,639,411,066,623
42	−406,373,387,911,744,607
44	518,422,953,957,560,676
46	−574,807,649,440,327,458
48	551,828,147,671,441,734
50	−456,602,133,188,515,554
52	323,842,004,554,570,254
54	−195,583,677,627,033,074
56	99,799,512,244,940,281
58	−42,623,325,522,793,081
60	15,065,907,534,807,099
62	−4,347,541,972,528,117
64	1,007,231,299,977,324
66	−183,492,571,858,506
68	25,601,107,765,586
70	−2,643,389,537,838
72	192,895,705,906
74	−9,324,974,318
76	271,097,321
78	−4,043,149
80	22,771
82	−21
Matching Polynomial of icosikaihenaacene: C86H46
Xn−k	Ck
0	1
2	−106
4	5399
6	−175,982
8	4,124,817
10	−74,063,694
12	1,059,798,817
14	−12,413,256,034
16	121,312,317,953
18	−1,003,323,241,558
20	7,098,532,567,167
22	−43,320,026,062,610
24	229,505,620,258,568
26	−1,060,826,821,769,124
28	4,294,355,311,546,214
30	−15,268,301,022,112,204
32	47,775,302,500,611,115
34	−131,736,351,086,317,702
36	320,322,373,771,219,109
38	−686,879,644,763,111,634
40	1,298,296,235,256,059,369
42	−2,160,812,435,654,172,406
44	3,161,718,641,030,940,101
46	−4,058,460,408,832,005,802
48	4,557,683,667,571,399,942
50	−4,462,813,454,863,667,224
52	3,794,792,897,456,339,076
54	−2,788,519,170,495,778,816
56	1,760,616,669,946,163,818
58	−948,642,540,487,223,764
60	432,694,958,770,892,042
62	−165,475,778,402,889,628
64	52,453,188,928,638,474
66	−13,591,558,098,242,156
68	2,830,490,059,458,396
70	−463,880,574,967,916
72	58,254,110,807,545
74	−5,414,880,348,970
76	355,639,745,651
78	−15,461,753,910
80	403,631,899
82	−5,390,418
84	27,049
86	−22

, containing up to 86 carbon atoms. It is readily seen that the number of perfect matchings for a D_2h linear system of polyacenes containing n rings is n+1, consistent with other combinatorial enumerations. This suggests a merely linear growth of the number of perfect matchings; hence, such a small growth is consistent with the decrease in the aromaticity of polyacenes (D_2h) with an increase in the number of hexagons. On the other hand, if the rings are fused in a kinked fashion as opposed to a linear fashion, the number of perfect matchings grows, for example, in fibonacenes as a Fibonacci series. For other partially kinked structures, the numbers grow in a Fibonacci-like series with an incremental add-on for each kink in the structure. This clearly explains the enhanced stability of the zigzag structures of fused benzene rings compared to the linear fashion of the D_2h structure in Figure 5.

F.

Matching Polynomials of Fan and Wheel Graphs

A group of related graphs called the fan and wheel graphs has been the topic of a number of studies over the decades due to their recursive nature and some interesting mathematical and spectral properties [90,91,92]. The fan graphs have also been studied in the context of matching generators and normal volumes of these graphs [19]. Although the matching polynomials of these structures have been explicitly constructed for smaller graphs [20], some of the coefficients previously reported are erroneous. As the matching polynomials of the C_n graphs are connected to the Lucas polynomials [19,92], the interplay between the Lucas polynomials and the matching polynomials of fan and wheel graphs has been of interest.

Figure 6 shows a typical fan graph and a wheel graph containing nine vertices. The wheel graph is also a variant representation of a pyramid formed by a cycle of length n with a single vertex being the apex of the pyramid. Both of these graphs are of interest due to the nature of their matching polynomials and their relationship to other well-characterized polynomials. In

Table 6. Matching polynomials of fan and wheel graphs.

Matching polynomial of fan graph:F1,8
Xn−k	Ck
0	1
2	−15
4	57
6	−70
8	21
Matching polynomial of fan graph:F1,9
Xn−k	Ck
0	1
2	−17
4	77
6	−125
8	65
10	−5
Matching polynomial of fan graph:F1,59
Xn−k	Ck
0	1
2	−117
4	4902
6	−115,500
8	1,810,215
10	−20,556,315
12	177,920,470
14	−1,212,786,120
16	6,657,068,250
18	−29,886,244,850
20	110,940,604,236
22	−343,090,279,272
24	888,273,814,467
26	−1,930,565,260,575
28	3,525,156,951,030
30	−5,402,923,000,816
32	6,932,619,043,842
34	−7,414,579,956,042
36	6,568,660,819,500
38	−4,779,873,142,200
40	2,825,853,840,810
42	−1,338,323,167,170
44	498,643,637,220
46	−142,773,786,000
48	30,457,490,700
50	−4,641,936,156
52	476,333,352
54	−30,105,712
56	1,011,375
58	−13,515
60	30
Matching polynomial of fan graph:F1,85
Xn−k	Ck
0	1
2	−169 (132)
4	10,375 (83 × 53)
6	−364,203 (33 × 7 × 41 × 47)
8	8,659,980 (22 × 33 × 5 × 7 × 29 × 79)
10	−152,147,996
12	2,080,963,885
14	−22,933,178,125
16	208,617,123,495
18	−1,594,369,711,175
20	10,374,198,391,655
22	−58,060,344,219,059
24	281,714,375,014,800
26	−1,192,427,220,013,008
28	4,424,260,086,335,960
30	−14,442,497,433,011,224
32	41,594,650,278,748,204
34	−105,895,397,831,661,900
36	238,617,629,780,678,148
38	−476,182,683,153,477,300
40	841,563,020,406,566,640
42	−1,316,390,172,750,258,480
44	1,820,371,878,792,109,125
46	−2,221,526,893,121,028,885
48	2,386,893,699,264,226,425
50	−2,251,111,074,386,846,217
52	1,856,642,490,120,368,853
54	−1,333,123,593,620,443,625
56	828,854,313,170,681,336
58	−443,360,306,892,355,160
60	202,479,302,558,751,372
62	−78,232,205,870,140,140
64	25,294,376,054,504,375
66	−6,754,075,658,931,055
68	1,465,618,299,382,825
70	−253,359,004,049,445
72	34,025,040,410,564
74	−3,436,353,389,300
76	249,909,853,193
78	−12,317,366,985
80	375,894,519 (19,383 × 19,393) (3 × 7 × 11 × 13 × 41 × 43 × 71)
82	−6,122,039 (part of integer sequence A006414 for nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.
84	39,775 (52 × 37 × 43)
86	−43
Matching polynomial of wheel graph:W18
Xn−k	Ck
0	1
2	−34
4	374
6	−1989
8	5797
10	−9537
12	8568
14	−3774
16	629
18	−17
Matching polynomial of wheel graph:W20
Xn−k	Ck
0	1
2	−38
4	475
6	−2945
8	10,374
10	−21,736
12	26,961
14	−18,810
16	6555
18	−874
20	19
Matching polynomial of wheel graph:W86
Xn−k	Ck
0	1
2	−170 = $\left[\begin{array}{c}6\\ 4\end{array}\right]$× C(2,1))
4	10,540: $\left[\begin{array}{c}6\\ 4\end{array}\right]$× S(6,2) × 22
6	−374,085: S(9,2) × 163 × 32
8	8,998,100: $\left[\begin{array}{c}6\\ 4\end{array}\right]$× 67 × 79 × 52 × 23
10	−159,976,817
12	2,214,762,550
14	−24,711,381,475
16	227,636,971,275
18	−1,762,087,833,500
20	11,615,083,138,744
22	−65,865,738,744,845
24	323,880,131,505,840
26	−1,389,589,368,548,360
28	5,227,087,399,521,200
30	−17,302,653,035,266,792
32	50,541,458,989,916,620
34	−130,532,286,306,783,240
36	298,446,755,493,160,560
38	−604,444,951,535,743,500
40	1,084,392,669,948,364,560
42	−1,722,273,249,878,853,525
44	2,418,792,979,176,641,550
46	−29,98,596,873,327,102,075
48	3,273,690,120,802,954,125
50	−3,137,999,806,448,287,872
52	2,631,194,427,017,021,040
54	−1,921,253,875,028,804,855
56	1,215,086,813,713,908,520
58	−66,1347,989,014,027,700
60	307,419,817,142,915,064
62	−120,935,396,967,175,700
64	39,824,665,609,048,775
66	−10,834,363,223,810,350
68	2,396,195,651,846,500
70	−422,340,034,401,579
72	57,851,679,198,620
74	−5,961,837,763,765
76	442,600,039,310
78	−22,278,193,575
80	694,637,867
82	−11,564,420 ($\left[\begin{array}{c}6\\ 4\end{array}\right]\times$43 × 113 × 7 × 22)
84	76,840 ($\left[\begin{array}{c}6\\ 4\end{array}\right]$× 113 × 23)
86	−85: $\left[\begin{array}{c}6\\ 4\end{array}\right]$

, we have shown the matching polynomials of fan and wheel graphs containing up to 86 vertices. We have also shown the relationship of some of the coefficients of the matching polynomials of these structures to the Stirling numbers of the first and second kinds. The Stirling numbers of the first and second kinds occur naturally in a number of combinatorial problems, particularly in the matching polynomials of fans and wheels. The Stirling number of the second kind, S(n,k), enumerates the number of ways to generate partitions of a set of n objects into k disjoint subsets. Thus, they are also related to the combinatorics of the enumeration of finite topologies [61,93]. As we are concerned with disjoint dimers or matchings in the computations of matching polynomials, the Stirling numbers find their place in the combinatorial interpretation of some of the coefficients of the matching polynomials. Moreover, as we discuss below, the matching polynomials of these graphs bear recursive relationships similar to several orthogonal polynomials. The coefficients of such orthogonal polynomials also bear a relationship to the Stirling numbers, thus providing further insights into the combinatorial origin of these coefficients. As seen from Table 6, the second from the last coefficient of the fan graph F₈₅¹ is 6,122,039, which is part of the integer sequence A006414, which enumerates the number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.

A recursive relation for the matching polynomials of the fan graph can be obtained by successive pruning of the edges originating from the singleton vertex to the vertex that is part of the path graph P_n. For example,

M(F_n¹) = M(F_n¹ − e₁) − M(P_n−1),

M(F_n¹ − e1) = M(F_n¹ − e₁ − e₂) − xM(P_n−2),

where P_n−1 is a path of length n − 1, and F_n¹ − e₁ is the fan graph F_n−1¹ with a single pendant vertex. The matching polynomials of path graphs can be readily obtained in terms of the Chebychev polynomials [60]. A similar recursion relation can be obtained from the wheel graph by choosing one edge at a time from the center of the wheel to the peripheral vertex.

M(W_n¹) = M(W_n¹ − e₁) − M(P_n−1),

Consequently, it is clear that the matching polynomials of fan and wheel graphs would finally be reduced to a sum of Chebychev polynomials of different lengths. Table 6 shows the matching polynomial of the wheel graphs thus computed for W₂₀ and W_86. It is also interesting that several of the coefficients of the matching polynomials of these recursive graphs are expressible as a combination of Stirling numbers of the first and second kinds and prime factors (See, Table 6). We note that the Stirling numbers appear in a number of combinatorial applications including the enumeration of finite topologies and Borel fields [61,93]. The last-2 and last-1 coefficients are directly related to the last coefficient of the matching polynomials of wheels (see Table 6).

The matching polynomials of graphs are intimately connected to several orthogonal polynomials. For example, the matching polynomials of complete graphs K_n are the well-known Hermite polynomials. The matching polynomials of the path and monocyclic graphs, P_n and C_n, generate Chebychev polynomials of the first and second kinds. A relation exists between the matching polynomials and Lucas polynomials. Other orthogonal polynomials, such as the Laguerre polynomials, are also connected to the matching polynomials of graphs. Several recursive graphs, such as fans, wheels, Bethe lattices, cactus graphs, and so forth, all yield interesting recursive patterns of matching polynomials as they can be derived by recursive tree-pruning algorithms.

4. Conclusions

In this study, we have considered the matching polynomials of a variety of graphs such as polyhedra, 4D and 5D structures, and symmetric, semisymmetric, double group, and several recursive graph structures. The matching polynomials of these structures were shown to manifest interesting combinatorial patterns. These polynomials would find applications in a variety of fields. Moreover, they could provide similarity measures and aromaticity measures when combined with other parameters. We have also shown the use of these polynomials in the context of dynamic stereochemistry, twisted topologies, relativistic quantum chemistry, and so forth. It is hoped that these developments will catalyze future applications in nanomaterials, nanotubes of different topologies, and metal organic frameworks. The matching polynomials in combination with spectral polynomials have recently been shown to provide accurate measures of aromaticity [2]. However, it is unlikely that matching polynomials alone would provide any insights into chirality, as matching polynomials of two chiral isomers are identical. Consequently, chirality cannot be contrasted by the matching polynomials. On the other hand, several experimental studies have shown the important applications of topological structures in photon propagation and control in the field of photonics, especially for topological insulators [94,95]. Such applications involving the confinement of photons to a given volume could benefit from topological indices. Such topological descriptors could be potentially derived from the matching polynomials considered here, as well as other polynomials discussed in previous studies, such as delta and spectral polynomials. These topics could be the subject matter of future research in this fascinating area connecting topology to the structural features of molecules and solids.