1. Introduction
Throughout this paper we are concerned with finite undirected connected simple graphs (networks). Let G=(V,E) be a graph with vertices labelled 1,2,…,n. The adjacency matrix A(G) of G is an n×n matrix with the (i,j)-entry equal to 1 if vertices i and j are adjacent and 0 otherwise. Suppose D(G)=diag(d1(G),d2(G),…,dn(G)) is the degree diagonal matrix of G, where di(G) is the degree of the vertex i, i=1,2,…,n. Let L(G)=D(G)-A(G) be called the Laplacian matrix of G. Then, the eigenvalues of A(G) and L(G) are called eigenvalues and Laplacian eigenvalues of G, respectively.
Given graphs G and H with vertex sets U and V, the Cartesian product G □ H of graphs G and H is a graph such that the vertex set of G □ H is the Cartesian product U×V; and any two vertices (u,u′) and (v,v′) are adjacent in G □ H if and only if either u=v and u′ is adjacent with v′ in H or u′=v′ and u is adjacent with v in G [1]. It is well known that many of the graphs (networks) operations can produce a great deal of novel types of graphs (networks), for example, Cartesian product of graphs, line graph, subdivision graph, and so on. The clique-inserted graph, denoted by C(G), is defined as a line graph of the subdivision graph S(G) [2, 3]. The subdivision graph of an r-regular graph is (r,2)-semiregular graph. Consequently, the clique-inserted graph of an r-regular graph is the line graph of an (r,2)-semiregular graph.
The resistance distances between vertices i and j, denoted by rij, are defined as the effective electrical resistance between them if each edge of G is replaced by a unit resistor [4]. A famous distance-based topological index, the Kirchhoff index Kf(G), is defined as the sum of resistance distances between all pairs of vertices in G; that is, Kf(G)=(1/2)∑i=1n∑j=1nrij(G), known as the Kirchhoff index of G [4]; recently, this classical index has also been interpreted as a measure of vulnerability of complex networks [5].
The Kirchhoff index attracted extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [6–13]. Besides, the Kirchhoff index also is a structure descriptor [14]. Unfortunately, it is rather hard to directly design some algorithms [15–17] to calculate resistance distances and the Kirchhoff indexes of graphs. So, many researchers investigated some special classes of graphs [18–21]. In addition, many efforts were also made to obtain the Kirchhoff index bounds for some graphs [17, 22]. Details on its theory can be found in recent papers [17, 22] and the references cited therein.
Motivated by the above results, we present the corresponding calculating formulae for the Kirchhoff index of L(Tm×n), S(Tm×n), T(Tm×n), and C(Tm×n) in this paper. The rest of this paper is organized as follows. Section 2 presents some underlying notations and preliminaries in our discussion. The proofs of our main results and some asymptotic behavior of Kirchhoff index are proposed in Sections 3 and 4, respectively.
2. Notations and Some Preliminaries
In this section, we introduced some basic properties which we need to use in the proofs of our main results. Suppose that Tm×n stands for the graphs Cm □ Cn for the convenience of description. It is trivial for m,n are 1, 2, without loss of generality, we discuss the situations for any positive integer m,n≥3.
Zhu et al. [15] and Gutman and Mohar [8] proved the relations between Kirchhoff index of a graph and Laplacian eigenvalues of the graph as follows.
Lemma 1 (see [8, 15]).
Let G be a connected graph with n≥2 vertices and let μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of graph G; then
(1)Kf(G)=n∑i=1n-11μi.
The line graph of a graph G, denoted by L(G), is the graph whose vertices correspond to the edges of G with two vertices of L(G) being adjacent if and only if the corresponding edges in G share a common vertex. The subdivision graph of a graph G, denoted by S(G), is the graph obtained by replacing every edge in G with a copy of P2 (path of length two). The total graph of a graph G, denoted by T(G), is the graph whose vertices correspond to the union of the set of vertices and edges of G, with two vertices of T(G) being adjacent if and only if the corresponding elements are adjacent or incident in G. Let PG(x) be the characteristic polynomial of the Laplacian matrix of a graph G; the following results were shown in [23].
Lemma 2 (see [23]).
Let G be an r-regular connected graph with n vertices and m edges; then(2)PL(G)(x)=(x-2r)m-nPG(x),PS(G)(x)=(-1)m(2-x)m-nPG(x(r+2-x)),PT(G)(x) =(-1)m(r+1-x)n(2r+2-x)m-nPG(x(r+2-x)r-x+1),
where PL(G)(x), PS(G)(x), and PT(G)(x) are the characteristic polynomials for the Laplacian matrix of graphs L(G), S(G), and T(G), respectively.
A bipartite graph G with a bipartition V(G)=(U,V) is called an (r,s)-semiregular graph if all vertices in U have degree r and all vertices in V have degree s. Apparently, the subdivision graph of an r-regular-graph G is (r,2)-semiregular graph.
Lemma 3 (see [24]).
Let G be an (r,s)-semiregular connected graph with n vertices. Then
(3)PL(G)(x)=(-1)n(x-(r+s))m-nPG(r+s-x),
where PL(G)(x) is the Laplacian characteristic polynomial of the line graph L(G) and m is the number of edges of G.
Lemma 4 (see [23]).
Let G be a connected simple r-regular graph with n vertices and m edges and let L(G) be the line graph of G. Then
(4)Kf(L(G))=r2Kf(G)+14n(m-n).
Lemma 5 (see [23]).
Let G be a connected simple r-regular graph with n≥2 vertices; then
(5)Kf(S(G))=(r+2)22Kf(G)+(r2-4)n2+4n8.
The following lemma gives an expression on Kf(T(G)) and Kf(G) of a regular graph G.
Lemma 6 (see [25]).
Let G be a r-regular connected graph with n vertices and m edges, and r≥2; then
(6)Kf(T(G))=n(r+2)(r+4)2(r+3)∑i=1n-11μi+3+r +(r+2)22(r+3)Kf(G)+n2(r2-4)8(r+1)+n2.
Lemma 7 presents the formula for calculating Kirchhoff index of Tm×n; in the following proof, some techniques in [26] are referred to.
Lemma 7 (see [26]).
For the toroidal networks Tm×n with any positive integer m,n≥3,
(7)Kf(Tm×n)=mn∑i=1m-1 ∑j=1n-114sin2(iπ/m)+4sin2(jπ/n) +nm3-m12+mn3-n12.
Proof.
Suppose the Laplacian eigenvalues of Cm and Cn are 4sin2(iπ/m) and 4sin2(jπ/n), i=0,1,…,m-1; j=0,1,…,n-1; then the Cartesian product G □ H and the Laplacian eigenvalues of L(cm□Cn) are
(8)4sin2iπm+4sin2jπn, i=0,1,…,m-1; j=0,1,…,n-1.
According to Lemma 1, the Kirchhoff index of the toroidal networks Kf(Tm×n) is
(9)Kf(Tm×n)=mn∑(i,j)∈A×B∖{(0,0)}14sin2(iπ/m)+4sin2(jπ/n),A={0,1,…,m-1}, B={0,1,…,n-1}=mn∑i=0m-1 ∑j=0n-114sin2(iπ/m)+4sin2(jπ/n),hhhhhhhhhhhhhhhhhW(i,j)≠(0,0)(10)=mn(∑i=1m-114sin2(iπ/m)+∑j=1n-114sin2(jπ/n)hhhhh+∑i=1m-1 ∑j=1n-114sin2(iπ/m)+4sin2(jπ/n))=n(m∑i=1m-114sin2(iπ/m))+m(n∑j=1n-114sin2(jπ/n))+mn∑i=1m-1 ∑j=1n-114sin2(iπ/m)+4sin2(jπ/n)=nKf(Cm)+mKf(Cn)+mn∑i=1m-1 ∑j=1n-114sin2(iπ/m)+4sin2(jπ/n)=mn∑i=1m-1 ∑j=1n-114sin2(iπ/m)+4sin2(jπ/n)+nm3-m12+mn3-n12.
Since Kf(Cn)=(n3-n)/12, (10) in the last line holds.
The following consequence was presented in [26]. Here we give a short proof.
Lemma 8 (see [26]).
For the toroidal networks Tm×n with any positive integer m,n≥3,
(11)limm→∞ limn→∞Kf(Tm×n)m2n2≈1.905.
Proof.
By virtue of (9), one can derive that
(12)Kf(Tm×n)mn=∑i=0m-1 ∑j=0n-114sin2(iπ/m)+4sin2(jπ/n),000000000000000000000000000 (i,j)≠(0,0).
Hence,
(13)limm→∞ limn→∞Kf(Tm×n)m2n2=limm→∞ limn→∞1mn∑i=0m-1 ∑j=0n-114-2cos(2πi/m)-2cos(2πj/n),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh(i,j)≠(0,0)=12π12π∫02π∫02πdxdy4-2cosx-2cosy≈1.905.
4. The Asymptotic Behavior of Related Kirchhoff Index
We explore the asymptotic behavior of Kirchhoff index for the investigated networks above as m,n tend to infinity. It is interesting and surprising that the quantity tends to a constant even though Kf(G)→∞, as m,n tend to infinity; that is,
(36)limm→∞ limn→∞Kf(G)m2n2=C, m,n⟶∞.
Moreover, one can employ the applications of analysis approach to obtain the explicit approximate values of Kirchhoff index for the related networks.
Theorem 14.
Let L(Tm×n) be line graphs of Tm×n with any positive integer m,n; then
(37)limm→∞ limn→∞Kf(L(Tm×n))m2n2≈4.060.
Proof.
According to (15) and the result of Lemma 8, we can derive that
(38)Kf(L(Tm×n))=2Kf(Tm×n)+m2n24=4.060m2n2.
Consequently,
(39)limm→∞ limn→∞Kf(L(Tm×n))m2n2≈4.060.
The result is equivalent to L(Tm×n) having asymptotic Kirchhoff index,
(40)Kf(L(Tm×n))≈4.060m2n2, m,n⟶∞.
Theorem 15.
Let S(Tm×n) be subdivision graph of Tm×n with any positive integer m,n; then
(41)limm→∞ limn→∞Kf(S(Tm×n))m2n2≈35.790.
Proof.
Similarly, according to (18), we can easily verify that
(42)Kf(S(Tm×n))=18Kf(Tm×n)+3m2n2+mn2=35.790m2n2.
Hence,
(43)limm→∞ limn→∞Kf(S(Tm×n))m2n2≈35.790.
Theorem 16.
Let T(Tm×n) be total graph of Tm×n with any positive integer m,n≥3; then
(44)limm→∞ limn→∞Kf(T(Tm×n))m2n2≈5.521.
Proof.
Consider the summation term ∑i=0m-1∑j=0n-1(1/(7+4sin2(iπ/m)+4sin2(jπ/n))).
Since
(45)limm→∞ limn→∞1m1n∑i=0m-1 ∑j=0n-117+4sin2(iπ/m)+4sin2(jπ/n)=limm→∞ limn→∞1m1n∑i=0m-1 ∑j=0n-1111-2cos(2πi/m)-2cos(2πj/n)=14π2∫02π∫02πdxdy11-2cosx-2cosy≈0.094.
The value in last line via the mathematic software MATLAB, which can obtain the result above.
Combining with (22), we can obtain that
(46)Kf(T(Tm×n)) =187Kf(Tm×n) +24mn7∑i=0m-1 ∑j=0n-117+4sin2(iπ/m)+4sin2(jπ/n) +3m2n210+mn2, (i,j)≠(0,0) ≈5.521m2n2.
So
(47)limm→∞ limn→∞Kf(T(Tm×n))m2n2≈5.521.
Theorem 17.
Let C(Tm×n) be clique-inserted graph of Tm×n with any positive integer m,n≥3; then
(48)limm→∞ limn→∞Kf(C(Tm×n))m2n2≈38.591.
Proof.
From the proof of Theorem 12, we know that
(49)Kf(C(Tm×n)) =4mn∑i=0m-1 ∑j=0n-113-5+2cos(2πi/m)+2cos(2πj/n) +4mn∑i=0m-1 ∑j=0n-113+5+2cos(2πi/m)+2cos(2πj/n) +53m2n2,
where the first summation i=0,1,…,m-1; j=0,1,…,n-1, and (i,j)≠(0,0).
As m,n tend to infinity, it follows from the first summation term:
(50)limm→∞ limn→∞1m1n∑i=0m-1 ∑j=0n-113-5+2cos(2πi/m)+2cos(2πj/n) =14π2∫02π∫02πdxdy3-5+2cosx+2cosy≈9.037.
Similarly, it holds from the second summation term when m,n tend to infinity,
(51)limm→∞ limn→∞1m1n∑i=0m-1 ∑j=0n-113+5+2cos(2πi/m)+2cos(2πj/n) =14π2∫02π∫02πdxdy3+5+2cosx+2cosy≈0.195.
Combining with the consequences of Theorem 12 and (50) and (51), it follows that
(52)limm→∞ limn→∞Kf(C(Tm×n))m2n2≈38.591.
Summing up, we complete the proof.