Open Access This article is
- freely available
Symmetry 2011, 3(3), 472-486; https://doi.org/10.3390/sym3030472
On Symmetry of Independence Polynomials
Department of Computer Science and Mathematics, Ariel University Center of Samaria, Kiryat HaMada, Ariel 40700, Israel
Department of Computer Science, Holon Institute of Technology, 52 Golomb Street, Holon 58102, Israel
Author to whom correspondence should be addressed.
Received: 27 April 2011; in revised form: 20 June 2011 / Accepted: 22 June 2011 / Published: 15 July 2011
An independent set in a graph is a set of pairwise non-adjacent vertices, and is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while is the cardinality of a maximum matching. If is the number of independent sets of size k in G, then , , is called the independence polynomial of G (Gutman and Harary, 1986). If for all , then is called symmetric (or palindromic). It is known that the graph , obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer , one can build a graph H, such that: G is a subgraph of H, is symmetric, and .
Keywords:independent set; independence polynomial; symmetric polynomial; palindromic polynomial
Classification:MSC 05C31; 05C69
Throughout this paper is a simple (i.e., a finite, undirected, loopless and without multiple edges) graph with vertex set and edge set If , then is the subgraph of G spanned by X. By we mean the subgraph , if . We also denote by the partial subgraph of G obtained by deleting the edges of F, for , and we write shortly , whenever F .
The neighborhood of a vertex is the set and , while ; if there is no ambiguity on G, we write and .
denote, respectively, the complete graph on vertices, the chordless path on vertices, and the chordless cycle on vertices.
The disjoint union of the graphs is the graph having as vertex set the disjoint union of , and as edge set the disjoint union of . In particular, denotes the disjoint union of copies of the graph G.
If are disjoint graphs, , then the Zykov sum of with respect to , is the graph with as vertex set andas edge set . If and , we simply write .
The corona of the graphs G and H with respect to is the graph obtained from G and copies of H, such that every vertex belonging to A is joined to all vertices of a copy of H . If we use instead of (see Figure 1 for an example).
Let be two graphs and C be a cycle on q vertices of G. By we mean the graph obtained from G and q copies of H, such that each two consecutive vertices on C are joined to all vertices of a copy of H (see Figure 2 for an example).
An independent (or a stable) set in G is a set of pairwise non-adjacent vertices. By we mean the family of all independent sets of G. An independent set of maximum size will be referred to as a maximum independent set of G, and the independence number of G, denoted by , is the cardinality of a maximum independent set in G.
Let be the number of independent sets of size k in a graph G. The polynomialis called the independence polynomial of G [3,4], the independent set polynomial of G . In , the dependence polynomial of a graph G is defined as .
A matching is a set of non-incident edges of a graph G, while is the cardinality of a maximum matching. Let be the number of matchings of size k in G.
The polynomialis called the matching polynomial of G .
The independence polynomial has been defined as a generalization of the matching polynomial, because the matching polynomial of a graph G and the independence polynomial of its line graph are identical. Recall that given a graph G, its line graph is the graph whose vertex set is the edge set of G, and two vertices are adjacent if they share an end in G. For instance, the graphs and depicted in Figure 3 satisfy and, hence, .
In  a number of general properties of the independence polynomial of a graph are presented. As examples, we mention that:
The following equalities are very useful in calculating of the independence polynomial for various families of graphs.
Let be a graph of order n. Then the following identities are true:
(i) holds for each .
(ii) for every graph H .
A finite sequence of real numbers is said to be:
- unimodal if there is some , such that ;
- log-concave if ;
- symmetric (or palindromic) if .
It is known that every log-concave sequence of positive numbers is also unimodal.
A polynomial is called unimodal (log-concave, symmetric) if the sequence of its coefficients is unimodal (log-concave, symmetric, respectively).
For instance, the independence polynomial:
- is log-concave;
- is unimodal, but it is not log-concave, because ;
- is non-unimodal;
- is symmetric and log-concave;
- is symmetric and non-unimodal.
It is easy to see that if and is symmetric, then it is also log-concave.
For other examples, see [9,10,11,12,13,14]. Alavi et al. proved that for every permutation of there is a graph G with such that .
The following conjecture is still open.
The independence polynomial of every tree is unimodal .
Hence to prove the unimodality of independence polynomials is sometimes a difficult task. Moreover, even if the independence polynomials of all the connected components of a graph G are unimodal, then is not for sure unimodal . The following result shows that symmetry gives a hand to unimodality.
If P and Q are both unimodal and symmetric, then is unimodal and symmetric .
A clique cover of a graph G is a spanning graph of G, each connected component of which is a clique. A cycle cover of a graph G is a spanning graph of G, each connected component of which is a vertex, an edge, or a proper cycle. In this paper we give an alternative proof for the fact that the polynomials , , and are symmetric for every clique cover , and every cycle cover of a graph G, where and are graphs built by Stevanović’s rules . Our main finding claims that the polynomial is divisible both by and .
The paper is organized as follows. Section 2 looks at previous results on symmetric independence polynomials, Section 3 presents our results connecting symmetric independence polynomials derived by Stevanović’s rules , while Section 4 is devoted to conclusions, future directions of research, and some open problems.
2. Related Work
The symmetry of the matching polynomial and the characteristic polynomial of a graph were examined in , while for the independence polynomial we quote [17,19,20]. Recall from  that G is called an equible graph if for some graph H. Both matching polynomials and characteristic polynomials of equible graphs are symmetric . Nevertheless, there are non-equible graphs whose matching polynomials and characteristic polynomials are symmetric.
It is worth mentioning that one can produce graphs with symmetric independence polynomials in different ways. For instance, the independence polynomial of the disjoint union of two graphs having symmetric independence polynomial is symmetric as well. Another basic graph operation preserving symmetry of the independence polynomial is the Zykov sum of two graphs with the same independence number. We summarize other constructions respecting symmetry of the independence polynomial in what follows.
2.1. Gutman’s Construction 
For integers , , let be the graph built in the following manner . Start with three complete graphs , and whose vertex sets are disjoint. Connect the vertex of with vertices of and with vertices of (see Figure 4 as an example).
The graph thus obtained has a unique maximum independent set of size three, and its independence polynomial is equal toHence the independence polynomial of iswhich is clearly symmetric and log-concave.
2.2. Bahls and Salazar’s Construction 
The -path of length is the graph with and . Such a graph consists of k copies of , each glued to the previous one by identifying certain prescribed subgraphs isomorphic to . Let be an integer. The d-augmented path is defined by introducing new vertices and edges . Let and be a subset of its vertices. Let and define the cone of G on U with vertex v, denoted , where . Given G and U and a graph H, we write instead of .
Let , and be integers, and let be a graph with a distinguished subset of vertices. Suppose that each of the graphs G, , and has a symmetric and unimodal independence polynomial, and . Then the independence polynomial of the graph is symmetric and unimodal .
2.3. Stevanović’s Constructions 
Taking into account that and , it follows that if is symmetric, then and , i.e., G has only one maximum independent set, say S, and independent sets, of size , that are not subsets of S.
If there is an independent set S in G such that holds for every independent set , then is symmetric .
The following result is a consequence of Theorem 2.2.
(i) If , and for the unique stability system S of G it is true that for each , then is symmetric ; (ii) If G is a claw-free graph with , then is symmetric.
Corollary 2.3 gives three different ways to construct graphs having symmetric independence polynomials .
- Rule 1. For a given graph G, define a new graph H as: .
For an example, see the graphs in Figure 5: , while
- A cycle cover of a graph G is a spanning graph of G, each connected component of which is a vertex (which we call a vertex-cycle), an edge (which we call an edge-cycle), or a proper cycle. Let be a cycle cover of G.
Rule 2. Construct a new graph H from G, denoted by , as follows: if is
(i) a vertex-cycle, say v, then add two vertices and join them to v;
(ii) an edge-cycle, say , then add two vertices and join them to both u and v;
(iii) a proper cycle, withthen add s vertices, say and each of them is joined to two consecutive vertices on C, as follows: is joined to , then is joined to , further is joined to , etc.
Figure 6 contains an example, namely, , while
- A clique cover of a graph G is a spanning graph of G, each connected component of which is a clique. Let be a clique cover of G.
Rule 3. Construct a new graph H from G, denoted by , as follows: for each , add two non-adjacent vertices and join them to all the vertices of Q.
Figure 7 contains an example, namely, , while
Let H be the graph obtained from a graph G according to one of the Rules 1, 2 or 3. Then H has a symmetric independence polynomial .
2.4. Inequalities and Equalities Following from Theorem 2.4
When inequalities connecting coefficients of the independence polynomial is under consideration, the symmetry mirrors the area, where they are already established. The following results illustrate this idea.
Let be with , and be the coefficients of . Then is symmetric, and 
Let H be a graph of order , Γ be a cycle cover of H that contains no vertex-cycles, G be obtained by Rule 2, and . Then is symmetric and its coefficients satisfy the subsequent inequalities 
Let , be the graphs obtained according to Rule 3 from , as one can see in Figure 8.
If , then 
(i) and , satisfies the following recursive relations:
(ii) is both symmetric and unimodal.
It was conjectured in  that is log-concave and has only real roots. This conjecture has been resolved as follows.
Let . Then 
(i) the independence polynomial of is
(ii) has only real zeros, and, therefore, it is log-concave and unimodal.
The following lemma goes from the well-known fact that the polynomial is symmetric if and only if it equals its reciprocal, i.e.,
Let , and be polynomials satisfying . If any two of them are symmetric, then the third is symmetric as well.
For , Theorem 1.1 givesSinceone can easily see that the polynomial satisfies the identity (1). Thus we conclude with the following.
For every graph G, the polynomial is symmetric .
3.1. Clique Covers Revisited
If A is a clique in a graph G, then for every graph H
Let and .
For , let us define the following families of independent sets:Since A is a clique, it follows that .
Case 1. .
In this case if and only if . Hence, for each size , we get that
Case 2. .
Now, every has for exactly one H, namely, the graph H whose vertices are joined to a. Hence, W may contain vertices only from .
On the other hand, each has for the unique H appearing in . Therefore, W may contain vertices only from .
Hence for each positive integer , we obtain that
Consequently, one may infer that for each size, the two graphs, and , have the same number of independent sets, in other words, .
Since has disjoint components identical to H, it follows that . ◊
If A is a clique in a graph G, then
If G is a graph of order n and Φ is a clique cover, then
Let . According to Corollary 3.4, each
(a) vertex-clique of yields as a factor of , since a vertex defines a clique of size 1;
(b) edge-clique of yields as a factor of , since an edge defines a clique of size 2 (see Figure 9 as an example);
(c) clique , produces as a factor of (see Figure 10 as an example).
Since the cliques of are pairwise vertex disjoint, one can apply Corollary 3.4 to all the q cliques one by one.
Using Corollary 3.4 and the fact that , we have
Repeating this process with , and taking into account that all the cliques of are pairwise disjoint, we obtainas required. ◊
Lemma 3.1 and Theorem 3.5 imply the following.
For every clique cover Φ of a graph G, the polynomial is symmetric .
Clearly, for every there exists a clique cover containing k non-trivial cliques, namely, edges. Consequently, we obtain the following.
For every graph G and for each non-negative integer , one can build a graph H, such that: G is a subgraph of H, is symmetric, and .
3.2. Cycle Covers Revisited
If C is a proper cycle in a graph G, then for every graph H
Let , , , and .
For an independent set , let us denote:
Case 1. .
In this case if an only if , since W is an arbitrary independent set of . Hence, for each size , we get that
Case 2. .
Then, we may assert thatsince W has to avoid all the “H-neighbors” of the vertices in , both in and .
Hence, for each positive integer , we get thatConsequently, one may infer that for each size, the two graphs, and , have the same number of independent sets. In other words, .
Since has disjoint components identical to H, it follows thatas required. ◊
If C is a proper cycle in a graph G, then
If G is a graph of order n and Γ is a cycle cover containing k vertex-cycles, then
According to Corollaries 3.4 and 3.9, each
(a) vertex-cycle of yields as a factor of , since each vertex defines a clique of size 1;
(b) edge-cycle of yields as a factor of , since every edge defines a clique of size 2;
(c) proper cycle produces as a factor (see Figure 11 as an example).
Let be a cycle cover containing k vertex-cycles, namely, .
Using Corollary 3.9 and the fact that , we have
Repeating this process with , and taking into account that all the cycles of are pairwise vertex disjoint, we obtainas claimed. ◊
Lemma 3.1 and Theorem 3.10 imply the following.
For every cycle cover Γ of a graph G, the polynomial is symmetric .
In this paper we have given algebraic proofs for the assertions in Theorem 2.4, due to Stevanović . In addition, we have shown that for every clique cover , and every cycle cover of a graph G, the polynomial is divisible both by and .
For instance, the graphs from Figure 12 have:, while
The characterization of graphs whose independence polynomials are symmetric is still an open problem .
Let us mention that there are non-isomorphic graphs with the same independence polynomial, symmetric or not. For instance, the graphs , , , presented in Figure 13 are non-isomorphic, while
Recall that a graph having at most two vertices with the same degree is called antiregular . It is known that for every positive integer there is a unique connected antiregular graph of order n, denoted by , and a unique non-connected antiregular graph of order n, namely . In  we showed that the independence polynomial of the antiregular graph is:
Let us mention that and , where denotes the complete bipartite graph on vertices. Notice that the coefficients of the polynomialsatisfy for , while , i.e., is “almost symmetric”.
Characterize graphs whose independence polynomials are almost symmetric.
It is known that the product of a polynomial and its reciprocal is a symmetric polynomial. Consequently, if and are reciprocal polynomials, then the independence polynomial of is symmetric, because .
Describe families of graphs whose independence polynomials are reciprocal.
We would like to thank one of the anonymous referees for helpful comments, which improved the presentation of our paper.
- Zykov, A.A. Fundamentals of Graph Theory; BCS Associates: Scottsdale, AZ, USA, 1990. [Google Scholar]
- Frucht, R.; Harary, F. On the corona of two graphs. Aequ. Math. 1970, 4, 322–325. [Google Scholar] [CrossRef]
- Gutman, I.; Harary, F. Generalizations of the matching polynomial. Utilitas Math. 1983, 24, 97–106. [Google Scholar]
- Arocha, J.L. Propriedades del polinomio independiente de un grafo. Rev. Cienc. Mat. 1984, V, 103–110. [Google Scholar]
- Hoede, C.; Li, X. Clique polynomials and independent set polynomials of graphs. Discrete Math. 1994, 125, 219–228. [Google Scholar] [CrossRef]
- Fisher, D.C.; Solow, A.E. Dependence polynomials. Discrete Math. 1990, 82, 251–258. [Google Scholar] [CrossRef]
- Farrell, E.J. Introduction to matching polynomials. J. Combin. Theory 1979, 27, 75–86. [Google Scholar] [CrossRef]
- Gutman, I. Independence vertex sets in some compound graphs. Publ. Inst. Math. 1992, 52, 5–9. [Google Scholar]
- Alavi, Y.; Malde, P.J.; Schwenk, A.J.; Erdös, P. The vertex independence sequence of a graph is not constrained. Congr. Numerantium 1987, 58, 15–23. [Google Scholar]
- Levit, V.E.; Mandrescu, E. On unimodality of independence polynomials of some well-covered trees. In LNCS 2731; Calude, C.S., Dinneen, M.J., Vajnovszki, V., Eds.; Springer: Berlin, Germany, 2003; pp. 237–256. [Google Scholar]
- Levit, V.E.; Mandrescu, E. A family of well-covered graphs with unimodal independence polynomials. Congr. Numerantium 2003, 165, 195–207. [Google Scholar]
- Levit, V.E.; Mandrescu, E. Very well-covered graphs with log-concave independence polynomials. Carpathian J. Math. 2004, 20, 73–80. [Google Scholar]
- Levit, V.E.; Mandrescu, E. Independence polynomials of well-covered graphs: Generic counterexamples for the unimodality conjecture. Eur. J. Comb. 2006, 27, 931–939. [Google Scholar] [CrossRef]
- Levit, V.E.; Mandrescu, E. The independence polynomial of a graph-a survey. In Proceedings of the 1st International Conference on Algebraic Informatics, Aristotle University of Thessaloniki, Greece, Thessaloniki, Greece, 2005; pp. 233–254. Available online: cai05.web.auth.gr/papers/20.pdf (accessed on 15 July 2011). [Google Scholar]
- Keilson, J.; Gerber, H. Some results for discrete unimodality. J. Am. Stat. Assoc. 1971, 334, 386–389. [Google Scholar] [CrossRef]
- Andrews, G.E. The Theory of Partitions; Addison-Wesley: Reading, Boston, MA, USA, 1976. [Google Scholar]
- Stevanović, D. Graphs with palindromic independence polynomial. Graph Theory Notes N. Y. Acad. Sci. 1998, XXXIV, 31–36. [Google Scholar]
- Kennedy, J.W. Palindromic graphs. Graph Theory Notes N. Y. Acad. Sci. 1992, XXII, 27–32. [Google Scholar]
- Gutman, I. A contribution to the study of palindromic graphs. Graph Theory Notes N. Y. Acad. Sci. 1993, XXIV, 51–56. [Google Scholar]
- Bahls, P.; Salazar, N. Symmetry and unimodality of independence polynomials of path-like graphs. Australas. J. Combin. 2010, 47, 165–176. [Google Scholar]
- Gutman, I. Independence vertex palindromic graphs. Graph Theory Notes N. Y. Acad. Sci. 1992, XXIII, 21–24. [Google Scholar]
- Levit, V.E.; Mandrescu, E. Graph operations and partial unimodality of independence polynomials. Congr. Numerantium 2008, 190, 21–31. [Google Scholar]
- Levit, V.E.; Mandrescu, E. A family of graphs whose independence polynomials are both palindromic and unimodal. Carpathian J. Math. 2007, 23, 108–116. [Google Scholar]
- Wang, Y.; Zhu, B.X. On the unimodality of independence polynomials of some graphs. Eur. J. Comb. 2011, 32, 10–20. [Google Scholar] [CrossRef]
- Merris, R. Graph Theory; Wiley-Interscience: New York, NY, USA, 2001. [Google Scholar]
- Behzad, M.; Chartrand, D.M. No graph is perfect. Am. Math. Mon. 1967, 74, 962–963. [Google Scholar] [CrossRef]
- Levit, V.E.; Mandrescu, E. On the independence polynomial of an antiregular graph. 2010. arXiv:1007.0880v1 [cs.DM]. Available online: arxiv.org/PS_cache/arxiv/pdf/1007/1007.0880v1.pdf (accessed on 15 July 2011).
Figure 1. and , where .
Figure 2. G and , where and .
Figure 3. is the line-graph of and .
Figure 4. and .
Figure 5. G and .
Figure 6. G and , where .
Figure 7. G and , where .
Figure 8. and .
Figure 9. , .
Figure 10. , , and .
Figure 11. , and
Figure 12. G with and .
Figure 13. Non-isomorphic graphs.
© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).