The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

For a graph G, the resistance distance rG(x, y) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R∗(G) = ∑{x,y}⊂V(G) dG(x)dG(y)rG(x, y), where dG(x) is the degree of vertex x, and V(G) denotes the vertex set of G. L. Feng et al. obtained the element in Cact(n; t) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R∗(G), and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.


Introduction
Throughout this paper, we consider finite, undirected simple graphs.Let G = (V(G), E(G)) be a graph with vertex set V(G) (or V) and edge set E(G).For a graph G, the distance between vertices x and y, denoted by d G (x, y), is the length of a shortest path between them.
For distance, Harold Wiener in 1947 defined a famous index W(G) [1], named Wiener index, where W(G) = ∑ x,y∈V d G (x, y).It is the earliest and one of the most thoroughly studied distance-based graph invariants.Later, Dobrynin and Kochetova [2] gave a modified version of the Wiener index D + (G) = ∑ x,y∈V (d G (x) + d G (y))d G (x, y).It is called degree distance and has attracted much attention (see [3][4][5][6]).For a graph G, the degree distance D + (G) is the essential part of the molecular topological index MTI(G) introduced by Schultz [7], which is defined as MTI(G) = ∑ x∈V d 2 G (x) + D + (G), where ∑ x∈V d 2 G (x) is the well-known first Zagreb index [8].Klein et al. [9] discovered the relation between degree distance and Wiener index for a tree G on n vertices: The Gutman index of a connected graph G is defined as D * (G) = ∑ x,y∈V d G (x)d G (y)d G (x, y).It was introduced in [10] and has been studied extensively (see, e.g., [11,12]).For a tree G on n vertices, Gutman [10] showed that In 1993, Klein and Randić [13] introduced a distance function named resistance distance on a graph.They viewed a graph G as an electrical network such that each edge of G is assumed to be a unit resistor, and the resistance distance between the vertices x and y of the graph G, denoted by r G (x, y), is then defined to be the effective resistance between the vertices x and y in G.The Kirchhoff index K f (G) of G is defined as x,y∈V r G (x, y).
The index has been widely studied in mathematical, physical and chemical aspects; for details on the Kirchhoff index, the readers are referred to [14][15][16][17][18].In 1996, Gutman and Mohar [19] obtained the result by which a relationship is established between the Kirchhoff index and the Laplacian spectrum: , where µ 1 ≥ µ 2 ≥ . . .≥ µ n = 0 are the eigenvalues of the Laplacian matrix of a connected graph G with n vertices.
Similarly, if the distance is replaced by resistance distance in the expression for the degree distance and Gutman index, respectively, then one arrives at the following indices R + (G) and R * (G) are called the additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index, respectively, and were introduced by Gutman et al. [20] and Chen et al. [21], respectively.The indices have been well studied in both mathematical and chemical literature.In [22] some properties of R + (G) are determined and the extremal graph of cacti with minimum R + -value characterized.Bianchi et al. [23] studied some upper and lower bounds for G + (G) whose expressions do not depend on the resistance distances.Feng et al. [24] characterized n-vertex unicyclic graphs having maximum, second maximum, minimum, and second minimum multiplicative degree-Kirchhoff index.Palacios [25] studied some interplay of the three Kirchhoff indices and found lower and upper bounds for the additive degree-Kirchhoff index.Yang and Klein [26] derived a formula for R * (G) of subdivisions and triangulations of graphs.To simplify the calculation of R * (G), the present authors [27] also obtained a formula for R * (G) with respect to the subgraph of G.For more work on the topological indices, we refer the reader to [13,21,22,[28][29][30][31].
In this paper, we study the multiplicative degree-Kirchhoff index of cacti.To state our results, we introduce some notation and terminology.For graph-theoretical terms that are not defined here, we refer to Bollobás' book [32].Let P n , C n and S n be the path, the cycle and the star on n vertices, respectively.We denote by A graph G is called a cactus if each block of G is either an edge or a cycle.Denote by Cact(n; t) the set of cacti possessing n vertices and t cycles.
are of degree two, and the degree of vertex v k is greater than two.The vertex v k ∈ V(C) is called the anchor of C. Let G 0 (n; t) be the graph shown in Figure 1.In this paper, we first give some transformations on R * (G), and then, by these transformations, we determine the first-minimum and second-minimum multiplicative degree-Kirchhoff index in Cact(n; t) and characterize the corresponding extremal graphs, respectively.Now, we give some lemmas that are used in the proof of our main results.Lemma 1. Ref. [13] Let u be a cut vertex of a connected graph G and x and y be vertices occurring in different components which arise upon deletion of u, then r G (x, y) = r G (x, u) + r G (u, y).Lemma 2. Ref. [27] Let G 1 and G 2 be connected graphs with disjoint vertex sets, with m 1 and m 2 edges, respectively.Let u Constructing the graph G by identifying the vertices u 1 and u 2 , and denote the so obtained vertex by u.Then, For completeness, we also give the proof in this paper.
, where U n is the class of unicyclic graphs.The equality holds if and only if T ∼ = G 0 (n; 1).

Transformations
In this section, we give some transformations that decrease R * (G).
Transformation 1.Let u 1 u 2 be a cut-edge of G, but not an pendent edge, G 1 , G 2 be the connected components of G − u 1 u 2 , where u 1 ∈ V(G 1 ), u 2 ∈ V(G 2 ).Constructing the graph G from G by deleting u 1 u 2 and identifying the vertices u 1 , u 2 , denote the so obtained vertex by u, adding an pendent edge uv (as shown in Figure 2).Lemma 4. Let G, G be the graphs described in Transformation 1, then R * (G) > R * (G ).
Let H be the graph obtained by attaching to the vertex Let G n be the class of connected graphs on n vertices.By Transformation 1 and Lemma 4, we have the following result.
Corollary 1.Let G 0 be a graph with the smallest multiplicative degree-Kirchhoff index in G n , then all cut-edges are pendent edges.
Transformation 2. For G ∈ Cact(n; t), let C k be a cycle with k(≥ 4) vertices, contained in G. Let there be a unique vertex u ∈ C h which is adjacent to a vertex in V(G) − V(C).Assuming that uv, vw ∈ E(C), construct a new graph G * = G − vw + uw (as shown in Figure 3).For u ∈ V(C k ), by direct calculation, we have Lemma 5. Let G, G * be the graphs described in Transformation 2, then R * (G) > R * (G * ).
Proof.Let S be the graph obtained by attaching to the vertex u of C k−1 the pendent vertex v.
By Lemma 1, we have d S (y)r S (u, y).
Further by Equations ( 1) and ( 2), then Transformation 3. Let G ∈ Cact(n; t), t ≥ 2, be a cactus without edges.Let C be an end cycle of G and u be its anchor.Let v be a vertex of C different from u.The graphs G 1 and G 2 are constructed by adding r pendent edges to the vertices u and v of G respectively (as shown in Figure 4).Lemma 6.Let G, G 1 , G 2 be the graphs described in Transformation 3, then R * (G 2 ) > R * (G 1 ).
By Lemma 2, we have This completes the proof.
Suppose that G is a cactus graph and u, v ∈ V(G) are two vertices, such that ux i y i u (i = 1, 2, . . ., s) and vp j q j v (j = 1, 2, . . ., t) are pendant triangles with the anchor u and v, respectively.We form two new graphs A and B according to the following transformation.
Lemma 7. Let G, A, B be the graphs described in Transformation 4, then either R * (G) Proof.Let M = {x 1 , y 1 , . . ., x s , y s }, N = {p 1 , q 1 , . . ., p t , q t } and and analogously Considering that r(u, y) = 2 3 for y ∈ M and r(x, y) = r(x, u) + r(u, y), we get and analogously, Because After the transformation, the degree of the vertex v increases by 2s, and By Equations ( 3)-( 13), we have Similarly, we have Further, we have This completes the proof.Similar to the proof of Lemma 7, we can prove the following result.

Main Results
In this section, we determine the elements in Cact(n; t) with first-minimum and second-minimum multiplicative degree-Kirchhoff index by the transformations that we have obtained.Note that the first-minimum multiplicative degree-Kirchhoff index has been obtained in [33]; for completeness, we also give the following proof.Theorem 1. Ref. [33] Let G ∈ Cact(n; t), then R * (G) 34  3 t(n − 2t − 1).The equality holds if and only if G ∼ = G 0 (n, t).
Proof.Let G be the unique graph having the minimum multiplicative degree-Kirchhoff index in Cact(n; t).Case 1.If t = 1, Cact(n; t) is the class of unicyclic graphs.By Lemma 3, we know the results hold.Case 2. If t = 2, Cact(n; t) is the class of bicyclic graphs.By Lemma 4, we conclude that G contains two cycles attached to a common vertex u, and all cut-edges are all pendent edges (if any).Further, by Lemmas 8 and 6, all pendent edges (if any) are also attached to u.Finally, by Lemma 5, the two cycles must be triangles, that is, G ∼ = G 0 (n, 2).This obtains the desirable results.
Case 3. If t ≥ 3, by Lemma 4, we conclude that all cut-edges are all pendent edges (if any) in G. Further, by Lemmas 8 and 6, G has at least two end cycles.Repeated by Lemmas 5-8, we arrive at the conclusion G ∼ = G 0 (n, t) By direct calculation, we have This completes the proof.
. The equality holds if and only if G ∼ = G 0 3 .
Proof.By Lemmas 4-8 and Theorem 1, one can conclude that G, which the second-minimum multiplicative degree-Kirchhoff index in Cact(n; t) must be one of the graphs G 0 1 , G 0 2 , G 0 3 , as shown in Figure 5.By Lemma 2, we have This completes the proof.By Theorems 1 and 2, we have Corollary 2. Among all graphs in Cact(n; t), G 0 (n; t) and G 0 3 are the graphs with first-minimum and second-minimum multiplicative degree-Kirchhoff index.
According to the above discussion, we find that the extremal cacti for the index R * (G) are the same as the extremal cacti for the Kirchhoff index, the multiplicative degree-Kirchhoff index, the Wiener index and the other indices [22,29,34,35].Based on the known results for these indices, we guess the element of Cact(n; t) with maximum multiplicative degree-Kirchhoff index is isomorphic to the graph C n,t (as shown in Figure 6).Conjecture 1.Let C n,t be the graph depicted in Figure 6, where k = t 2 .Then, C n,t is the unique element of Cact(n; t) having maximum multiplicative degree-Kirchhoff index.
In particular, for Cat(n; t), if t = 1; 0, Cat(n; 1) and Cat(n; 0) are the set of unicyclic graphs and trees, respectively.For G ∈ Cat(n; 1), the graphs having maximum and minimum multiplicative degree-Kirchhoff index are given in [13], that is 3 ).
where U n 3 consists of a cycle of size 3 to which a path with n − 3 vertices is attached.For G ∈ Cat(n; 0), it is easy to get the result R * (S n ) ≤ R * (G) ≤ R * (P n ).

Conclusions
In this paper, we give some transformations on the multiplicative degree-Kirchhoff index.As applications, the second-minimum multiplicative degree-Kirchhoff index on Cat(n; t) and the corresponding extremal graph are determined.We guess C n,t is the graph of Cat(n; t) with maximum R * (G) value.For solving the problem, our approach would need to be modified; it would be interesting to continue studying the extremal graphs.
we denote by G − E 0 the subgraph of G obtained by deleting the edges in E 0 .If E 1 is the subset of the edge set of the complement of G, G + E 1 denotes the graph obtained from G by adding the edges in E 1 .Similarly, if W ⊂ V(G), we denote by G − W the subgraph of G obtained by deleting the vertices of W and the edges incident with them and G[W] the subgraph of G induced by W. If E = {xy} and W = {x}, we write G − xy and G − x instead of G − {xy} and G − {x}, respectively.

Figure 2 .
Figure 2. The graphs G and G in Transformation 1.

Figure 3 .
Figure 3.The graphs G and G * in Transformation 2.

Figure 4 .
Figure 4.The graphs G, G 1 and G 3 in Transformation 3.

Figure 6 .
Figure 6.The graphs C n,t .