Generalized Characteristic Polynomials of Join Graphs and Their Applications

Abstract

The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G in electrical networks. LEL(G) is the Laplacian-Energy-Like Invariant of G in chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex join [subscript]G1[/subscript] [ecedil]a;[subscript]G2[/subscript] and the subdivision-edge-edge join [subscript]G1[/subscript] [ecedil]d;[subscript]G2[/subscript] . We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials of [subscript]G1[/subscript] [ecedil]a;[subscript]G2[/subscript] and [subscript]G1[/subscript] [ecedil]d;[subscript]G2[/subscript] when [subscript]G1[/subscript] is [subscript]r1[/subscript] -regular graph and [subscript]G2[/subscript] is [subscript]r2[/subscript] -regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, and LEL of [subscript]G1[/subscript] [ecedil]a;[subscript]G2[/subscript] and [subscript]G1[/subscript] [ecedil]d;[subscript]G2[/subscript] in terms of the Laplacian spectra of [subscript]G1[/subscript] and [subscript]G2[/subscript] .

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Pengli Lu 1 and Ke Gao 1 and Yang Yang 1

Academic Editor:Francisco R. Villatoro

School of Computer and Communication, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 11 November 2016; Revised 18 January 2017; Accepted 26 January 2017; 2 March 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let G=(V(G),E(G)) be a simple graph on n vertices and m edges. Let _di =_dG (_vi ) be the degree of vertex _vi in G and D(G) be the diagonal matrix with diagonal entries _d1 ,_d2 ,...,_dn . Let A(G) denote the adjacency matrix of a graph G. The Laplacian matrix and the signless Laplacian matrix of G are defined as L(G)=D(G)-A(G) and Q(G)=D(G)+A(G), respectively. Let ψ(A(G);x)=det[...](x_In -A(G)), or simply ψ(A(G)) (ψ(L(G)) and ψ(Q(G)), resp.), be the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomial of G and its roots be the adjacency (Laplacian and signless Laplacian, resp.) eigenvalues of G, denoted by _λ1 (G)≥_λ2 (G)≥[...]≥_λn (G) (0=_μ1 (G)<=_μ2 (G)<=[...]<=_μn (G) and _ν1 (G)<=_ν2 (G)<=[...]<=_νn (G), resp.). The line graph of G is denoted by l(G).

The generalized characteristic polynomial of G, introduced by Cvetkovi et al. [1], is defined to be _[varphi]G (x,t)=det[...](x_In -(A(G)-tD(G))). The generalized characteristic polynomial covers the cases of usual characteristic polynomial A(G) and Laplacian L(G) and signless Laplacian Q(G) polynomials of graph G, due to variation of the parameter t and, whenever necessary, replacing x by -x. We can get that the characteristic polynomials of A(G), L(G), and Q(G) are equal to _[varphi]G (x,0), (-1^)|V(G)|_[varphi]G (-x,1), and _[varphi]G (x,-1).

Let G be a connected graph. For two vertices u and v of G, the resistance distance between u and v is defined to be the effective resistance between them when unit resistors are placed on every edge of G. It is a distance function on graphs introduced by Klein and Randic [2]. The Kirchhoff index of G, denoted by Kf(G), is the sum of resistance distances between all pairs of vertices of G. For a connected graph G of order n [3], Kf(G)=n^∑i=2n (1/_μi ). Recently, many results on Kirchhoff index were obtained [2, 4-8]. Laplacian-Energy-Like Invariant LEL(G)=^∑i=2n_μi was named in [9]. The motivation for introducing LEL was in its analogy to the earlier studied graph energy and Laplacian energy [10]. Although Kirchhoff index and LEL both depend on Laplacian eigenvalues, their comparative study started only quite recently [11, 12].

Graph operations, such as the disjoint union , the join , the corona , the edge corona , and the neighborhood corona [13-17], are techniques to construct new classes of graphs from old ones. In [18], a real molecular graph of ferrocene is a join graph obtained from graphs _G1 =_K1 and _G2 =2_C5 , where 2_C5 is a disjoint union of two pentagons. In [4-6, 8, 14] the resistance distance and Kirchhoff index of artificial graphs are computed. Although most of the constructed graphs in the literature are contrived, they may be of use for chemical and physical applications.

Motivated by the work above, we define two new graph operations based on subdivision graphs as follows.

The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G [19]. We denote the set of such new vertices by I(G). In [15, 20], some new graph operations based on subdivision graphs were defined and the A-, L-, and Q-spectrum were computed in terms of those of the two graphs.

Let _G1 and _G2 be two vertex disjoint graphs. The subdivision-vertex-vertex join of _G1 and _G2 , denoted by _G1 [ecedil]a;_G2 , is the graph obtained from S(_G1 ) and S(_G2 ) by joining every vertex in V(_G1 ) to every vertex in V(_G2 ). The subdivision-edge-edge join of _G1 and _G2 , denoted by _G1 [ecedil]d;_G2 , is the graph obtained from S(_G1 ) and S(_G2 ) by joining every vertex of I(_G1 ) to every vertex in I(_G2 ).

Note that if _G1 is a graph on _n1 vertices and _m1 edges and _G2 is a graph on _n2 vertices and _m2 edges, then _G1 [ecedil]a;_G2 has _n1 +_m1 +_n2 +_m2 vertices and 2_m1 +2_m2 +_n1n2 edges and _G1 [ecedil]d;_G2 has _n1 +_m1 +_n2 +_m2 vertices and 2_m1 +2_m2 +_m1m2 edges.

Let _Cn denote a cycle of order n and _Pm denote a path of order m. Figure 1 depicts the subdivision-vertex-vertex join _C5 [ecedil]a;_P2 and subdivision-edge-edge join _C5 [ecedil]d;_P2 , respectively.

Figure 1: An example of subdivision-vertex-vertex join and subdivision-edge-edge join .

[figure omitted; refer to PDF]

The paper is organized as follows. In Section 2, some useful lemmas are provided. In Section 3, we compute the generalized characteristic polynomial of the subdivision-vertex-vertex join and obtain the A-, L-, and Q-spectrum in terms of the corresponding spectrum of _G1 and _G2 when _G1 and _G2 are regular graphs. In Section 4, we compute the generalized characteristic polynomial of the subdivision-edge-edge join and obtain the A-, L-, and Q-spectrum in terms of the corresponding spectrum of _G1 and _G2 when _G1 and _G2 are regular graphs. In Section 5, as the applications, the number of spanning trees, Kirchhoff index, and LEL of the subdivision-vertex-vertex join and the subdivision-edge-edge join graphs are computed. In Section 6, we conclude the paper.

2. Preliminaries

The M-coronal _ΓM (x) of an n×n matrix M is defined [16] to be the sum of entries of the matrix (x_In -M^)-1 ; that is [figure omitted; refer to PDF] where _1n denotes the column vector of size n when all the entries are equal to one.

Lemma 1 (see [21]).

Let A be an n×n real matrix and let _Js×t denote the s×t matrix with all entries equal to one. Then [figure omitted; refer to PDF] where α is a real number and adj(A) is the adjugate matrix of A. Moreover, [figure omitted; refer to PDF]

Lemma 2 (see [13]).

If M is an n×n matrix when each row sum is equal to a constant t; then, [figure omitted; refer to PDF]

Lemma 3 (see [22]).

Let _M1 , _M2 , _M3 , and _M4 be, respectively, p×p, p×q, q×p, and q×q matrices with _M1 and _M4 invertible. Then [figure omitted; refer to PDF] where _M1 -_M2^M4-1_M3 and _M4 -_M3^M1-1_M2 are called the Schur complements of _M4 and _M1 , respectively.

Lemma 4 (see [19]).

If G is a regular graph of degree r, with n vertices and m edges; then [figure omitted; refer to PDF]

If A=[_aij ] is an m×n matrix and B is an r×s matrix, then the Kronecker product A[ecedil]7;B is defined as the mr×ns matrix with the block form [figure omitted; refer to PDF]

This is an associative operation with the property that ^{(A[ecedil]7;B)T} =^AT [ecedil]7;^BT and (A[ecedil]7;B)(C[ecedil]7;D)=AC[ecedil]7;BD whenever the products AC and BD exist. The latter implies ^{(A[ecedil]7;B)-1} =^A-1 [ecedil]7;^B-1 for nonsingular matrices A and B. Moreover, if A and B are n×n and p×p matrices, then det[...](A[ecedil]7;B)=^{(det[...]A)p(det[...]B)n} . The reader is referred to [23] for other properties of the Kronecker product not mentioned here.

3. Spectrum of Subdivision-Vertex-Vertex Join

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges, respectively. We first label the vertices of _G1 [ecedil]a; _G2 as follows. Let V(_G1 )=(_v1 ,_v2 ,...,_vn1 ), I(_G1 )=(_e1 ,_e2 ,...,_em1 ), V(_G2 )=(_u1 ,_u2 ,...,_un2 ), and I(_G2 )=(_e1 ,_e2 ,...,_em2 ). The adjacency matrix of _G1 [ecedil]a;_G2 can be written as follows: [figure omitted; refer to PDF] where _0s×t denotes the s×t matrix with all entries equal to zero, R is the incidence matrix of G, _In is the identity matrix of order n, and _1n is the column vector with all entries equal to 1. It is clear that the degrees of the vertices of _G1 [ecedil]a;_G2 are _{dG1 [ecedil]a; G2} (_vi )=_r1 +_n2 for i=1,2,...,_n1 , _{dG1 [ecedil]9; G2} (_ei )=2 for i=1,2,...,_m1 , _{dG1 [ecedil]a; G2} (_ui )=_r2 +_n1 for i=1,2,...,_n2 , and _{dG1 [ecedil]a; G2} (_ei )=2 for i=1,2,...,_m2 . Then the degree matrix of subdivision-vertex-vertex join can be written as follows: [figure omitted; refer to PDF]

3.1. Generalized Characteristic Polynomial of _G1 [ecedil]a;_G2

Theorem 5.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 be an _r2 -regular graph on _n2 vertices and _m2 edges. Then the generalized characteristic polynomial of subdivision-vertex-vertex join of _G1 and _G2 is [figure omitted; refer to PDF]

Proof.

Let _R1 be the incidence matrix of _G1 . Then, with respect to the adjacent matrix and degree matrix of _G1 [ecedil]a;_G2 , the generalized matrix of _G1 [ecedil]a;_G2 is given by [figure omitted; refer to PDF] Thus the generalized characteristic polynomial of _G1 [ecedil]a;_G2 is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Schur complement of (x+2t)_Im1 . So, det[...](S)=^(x+2t)_m2 det[...](_S1 ), where [figure omitted; refer to PDF] is the Schur complement of (x+2t)_Im2 . So, [figure omitted; refer to PDF] is the Schur complement of (x+_r1 t+_n2 t)_In1 -_R1^R1T /(x+2t). So, [figure omitted; refer to PDF] by Lemma 1, [figure omitted; refer to PDF]

Theorem 6.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges. Then

(1) the adjacency characteristic polynomial of _G1 [ecedil]a;_G2 is [figure omitted; refer to PDF]

(2) the Laplacian characteristic polynomial of _G1 [ecedil]a;_G2 is [figure omitted; refer to PDF]

(3) the signless Laplacian characteristic polynomial of _G1 [ecedil]a;_G2 is [figure omitted; refer to PDF]

4. Spectrum of Subdivision-Edge-Edge Join

The adjacency matrix of _G1 [ecedil]d;_G2 can be written as follows: [figure omitted; refer to PDF] where _0s×t denotes the s×t matrix with all entries equal to zero, R is the incidence matrix of G, _In is the identity matrix of order n, and _1n is the column vector with all entries equal to 1. It is clear that the degrees of the vertices of _G1 [ecedil]d;_G2 are _{dG1 [ecedil]d; G2} (_vi )=_r1 for i=1,2,...,_n1 , _{dG1 [ecedil]d; G2} (_ei )=_m2 +2 for i=1,2,...,_m1 , _{dG1 [ecedil]d; G2} (_ui )=_r2 for i=1,2,...,_n2 , and _{dG1 [ecedil]d; G2} (_ei )=_m1 +2 for i=1,2,...,_m2 . Then the degree matrix of subdivision-edge-edge join can be written as follows: [figure omitted; refer to PDF]

4.1. Generalized Characteristic Polynomial of _G1 [ecedil]d;_G2

Theorem 7.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 be an _r2 -regular graph on _n2 vertices and _m2 edges. Then the generalized characteristic polynomial of subdivision-edge-edge join of _G1 and _G2 is [figure omitted; refer to PDF]

Proof.

Let _R1 be the incidence matrix of _G1 . Then, with respect to the adjacent matrix and degree matrix of _G1 [ecedil]d;_G2 , the generalized matrix of _G1 [ecedil]d;_G2 is given by [figure omitted; refer to PDF] Thus the generalized characteristic polynomial of _G1 [ecedil]d;_G2 is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Schur complement of (x+_r1 t)_In1 . So, det[...](S)=^{(x+_r2 t)_n2} det[...](_S1 ), where [figure omitted; refer to PDF] is the Schur complement of (x+_r2 t)_In2 . So, [figure omitted; refer to PDF] is the Schur complement of (x+_m2 t+2t)_Im1 -^R1T_R1 /(x+_r1 t). So, [figure omitted; refer to PDF] by Lemma 4, [figure omitted; refer to PDF] and by Lemma 1, [figure omitted; refer to PDF]

Theorem 8.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges. Then

(1) the adjacency characteristic polynomial of _G1 [ecedil]d;_G2 is [figure omitted; refer to PDF]

(2) the Laplacian characteristic polynomial of _G1 [ecedil]d;_G2 is [figure omitted; refer to PDF]

(3) the signless Laplacian characteristic polynomial of _G1 [ecedil]d;_G2 is [figure omitted; refer to PDF]

5. Applications

We give the number of spanning trees, the Kirchhoff index, and LEL of the two classes of join graphs, as the applications.

We can easily compute the L-spectrum of _G1 [ecedil]a; _G2 in terms of L-spectrum of _G1 and _G2 by (2) of Theorem 6.

Corollary 9.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges. Then the Laplacian spectrum of _G1 [ecedil]a;_G2 consists of

(1) 2, repeated _m1 +_m2 -_n1 -_n2 times;

(2) two roots of the equation ^x2 -(_r1 +_n2 +2)x+2_n2 +_μi (_G1 )=0, for each i=2,...,_n1 ;

(3) two roots of the equation ^x2 -(_r2 +_n1 +2)x+2_n1 +_μi (_G2 )=0, for each i=2,...,_n2 ;

(4) four roots of the equation x(^x3 -(_n1 +_n2 +_r1 +_r2 +4)^x2 + (4_n1 +4_n2 +2_r1 +2_r2 +4+_r1r2 + _r1n1 +_r2n2 )x-4_n1 -4_n2 -2_n1r1 -2_n2r2 )=0, and the four roots are 0, _ω1 , _ω2 , and _ω3 , respectively.

We can easily compute the L-spectrum of _G1 [ecedil]d; _G2 in terms of L-spectrum of _G1 and _G2 by (2) of Theorem 8.

Corollary 10.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges. Then the Laplacian spectrum of _G1 [ecedil]d;_G2 consists of

(1) _m1 +2, repeated _m2 -_n2 times;

(2) _m2 +2, repeated _m1 -_n1 times;

(3) two roots of the equation ^x2 -(_m2 +2+_r1 )x+_r1m2 +_μi (_G1 )=0, for each i=2,...,_n1 ;

(4) two roots of the equation ^x2 -(_m1 +2+_r2 )x+_r2m1 +_μi (_G2 )=0, for each i=2,...,_n2 ;

(5) four roots of the equation x(^x3 -(_m1 +_m2 +_r1 +_r2 +4)^x2 + (_r1m2 +_r2m1 +_r2m2 +_r1m1 +_r1r2 + 2_m1 +2_m2 +2_r1 +2_r2 +4)x-_m1r1r2 -_m2r1r2 -2_m1r2 -2_m2r1 )=0, and the four roots are 0, _γ1 , _γ2 , and _γ3 , respectively.

Let t(G) denote the number of spanning trees [1] of G. It is well known that if G is a connected graph on n vertices with Laplacian spectrum 0=_μ1 (G)<=_μ2 (G)<=[...]<=_μn (G), then t(G)=(_μ2 (G)[...]_μn (G))/n.

Corollary 11.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges. The number of spanning trees of _G1 [ecedil]a; _G2 and _G1 [ecedil]d; _G2 are, respectively, [figure omitted; refer to PDF]

Proof.

For _G1 [ecedil]a;_G2 , the number of the vertices is _n1 +_m1 +_n2 +_m2 . By Corollary 9 and Vieta's formulas, we can obtain the following: for every _μi (_G1 ), ^∏i=2_n1 (_x1x2 )=^∏i=2_n1 (2_n2 +_μi (_G1 )); for every _μi (_G2 ), ^∏i=2_n2 (_x1x2 )=^∏i=2_n2 (2_n1 +_μi (_G2 )); for the equation ^x3 -(_n1 +_n2 +_r1 +_r2 +4)^x2 + (4_n1 +4_n2 +2_r1 +2_r2 +4+_r1r2 +_r1n1 +_r2n2 )x-4_n1 -4_n2 -2_n1r1 -2_n2r2 =0, _ω1ω2ω3 =4_n1 +4_n2 +2_r1n1 +2_r2n2 . Therefore, we complete the proof of t(_G1 [ecedil]a;_G2 ) and we omit the proof of t(_G1 [ecedil]d;_G2 ).

The Kirchhoff index [24] Kf(G) is an important physical and chemical indicator. Let G be a connected graph of order n; then Kf(G)=n^∑i=2n (1/_μi ).

Corollary 12.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges; then the Kirchhoff index of _G1 [ecedil]a; _G2 and _G1 [ecedil]d; _G2 is, respectively, [figure omitted; refer to PDF]

Proof.

For _G1 [ecedil]a;_G2 , by Corollary 9 and Vieta's formulas, we can obtain the following: for every _μi (_G1 ), ^∑i=2_n1 (1/_x1 +1/_x2 )=^∑i=2_n1 [...]((_x1 +_x2 )/_x1x2 )=^∑i=2_n1 [...]((_r1 +_n2 +2)/(2_n2 +_μi (_G1 ))); for every _μi (_G2 ), ^∑i=2_n2 (1/_x1 +1/_x2 )=^∑i=2_n2 ((_r2 +_n1 +2)/(2_n1 +_μi (_G2 ))); for the equation ^x3 -(_n1 +_n2 + _r1 +_r2 +4)^x2 + (4_n1 +4_n2 +2_r1 +2_r2 +4+_r1r2 +_r1n1 +_r2n2 )x-4_n1 -4_n2 -2_n1r1 -2_n2r2 =0, 1/_ω1 +1/_ω2 +1/_ω3 = (_ω1ω2 +_ω2ω3 +_ω1ω3 )/_ω1ω2ω3 = (4_n1 +4_n2 +2_r1 +2_r2 +4+_r1r2 +_r1n1 +_r2n2 )/(4_n1 +4_n2 +2_r1n1 +2_r2n2 ). Therefore, we complete the proof of Kf(_G1 [ecedil]a;_G2 ) and we omit the proof of Kf(_G1 [ecedil]d;_G2 ).

L E L ( G ) denotes Laplacian-Energy-Like Invariant [24]. Let G be a connected graph of order n with m edges; then LEL=LEL(G)=^∑i=2n_μi .

Corollary 13.

Let _G1 be an _r1 -regular graph on _n1 vertices and _m1 edges and _G2 an _r2 -regular graph on _n2 vertices and _m2 edges; then [figure omitted; refer to PDF] where _Δ1 =^{(_r1 +_n2 +2)2} -4(2_n2 +_μi (_G1 )), _Δ2 =^{(_r2 +_n1 +2)2} -4(2_n1 +_μi (_G2 )), and _ω1 , _ω2 , and _ω3 are three roots of the equation ^x3 -(_n1 +_n2 +_r1 +_r2 +4)^x2 + (4_n1 +4_n2 +2_r1 +2_r2 +4+_r1r2 + _r1n1 +_r2n2 )x-4_n1 -4_n2 -2_n1r1 -2_n2r2 =0. [figure omitted; refer to PDF] where _Δ3 =^{(_m2 +2+_r1 )2} -4(_r1m2 +_μi (_G1 )), _Δ4 =^{(_m1 +2+_r2 )2} -4(_r2m1 +_μi (_G2 )), and _γ1 , _γ2 , and _γ3 are three roots of the equation ^x3 -(_m1 +_m2 +_r1 +_r2 +4)^x2 + (_r1m2 +_r2m1 +_r2m2 +_r1m1 +_r1r2 +2_m1 + 2_m2 +2_r1 +2_r2 +4)x-_m1r1r2 -_m2r1r2 - 2_m1r2 -2_m2r1 =0.

6. Conclusions

In this paper, two classes of join graphs, _G1 [ecedil]a; _G2 and _G1 [ecedil]d; _G2 , are constructed by graph operations. The formulas of generalized characteristic polynomial of the two graphs are obtained by using block matrix and Schur complement. For regular graphs _G1 and _G2 , we characterize the A-, L-, and Q-spectrum of _G1 [ecedil]a; _G2 and _G1 [ecedil]d; _G2 in terms of the corresponding spectrum of _G1 and _G2 , and by the L-spectrum, we get the number of spanning trees, Kirchhoff index, and LEL. But for nonregular graphs _G1 and _G2 , we cannot determine the spectrum of _G1 [ecedil]a;_G2 and _G1 [ecedil]d;_G2 . The most difficult problem is that we cannot find the rule of the block matrices to deduce a simple result. Therefore, we should try to find new method to solve the problem.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 11361033).

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