Resistance Distance and Kirchhoff Index of Graphs with Pockets

Let G[F, Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1, Vk = {v1, v2, · · · , vk} is a subset of the vertex set of F and Hv is a simple graph of order m ≥ 2, v is a specified vertex of Hv. Also let G[F, Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek = {e1, e2, · · · ek} is a subset of the edge set of F and Huv is a simple graph of order m ≥ 3, uv is a specified edge of Huv such that Huv − u is isomorphic to Huv − v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F, Vk, Hv] and G[F, Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.


Introduction
All graphs considered in this paper are simple and undirected.The resistance distance between vertices u and v of G was defined by Klein and Randi ć [1] to be the effective resistance between nodes u and v as computed with Ohm's law when all the edges of G are considered to be unit resistors.The Kirchhoff index K f (G) was defined in [1] as K f (G) = ∑ u<v r uv , where r uv (G) denote the resistance distance between u and v in G. Resistance distance are, in fact, intrinsic to the graph, with some nice purely mathematical interpretations and other interpretations.The Kirchhoff index was introduced in chemistry as a better alternative to other parameters used for discriminating different molecules with similar shapes and structures [1].The resistance distance and the Kirchhoff index have attracted extensive attention due to its wide applications in physics, chemistry and others.Up till now, many results on the resistance distance and the Kirchhoff index are obtained.See ( [2], [3], [4], [5]) and the references therein to know more.However, the resistance distance and Kirchhoff index of the graph is, in general, a difficult thing from the computational point of view.Therefore, the bigger is the graph, the more difficult is to compute the resistance distance and Kirchhoff index, so a common strategy is to consider complex graph as composite graph, and to find relations between the resistance distance and Kirchhoff indices of the original graphs.
Let G = (V(G), E(G)) be a graph with vertex set V(G) and edge set E(G).Let d i be the degree of vertex i in G and ) the diagonal matrix with all vertex degrees of G as its diagonal entries.For a graph G, let A G and B G denote the adjacency matrix and vertex-edge incidence matrix of G, respectively.The matrix For other undefined notations and terminology from graph theory, the readers may refer to [6] and the references therein ([7]- [10]).
Recently Barik [11] described the Laplacian spectrum of graph with pockets except n + k Laplacian eigenvalues.Nath and Paul [12] gave the Laplacian spectrum of graph with edge pockets except n + k Laplacian eigenvalues.Tian [13] gave the spectrum and signless Laplacian spectrum of graph with pockets and edge pockets except n + k signless Laplacian eigenvalues.This paper considers the resistance distance and Kirchhoff index of graphs with pockets and edge pockets of these two new graph operations below, which come from [11] and [12], respectively.
Definition 1 [11] Let F, H v be graphs of orders n and m, respectively, where m ≥ 2, v be a specified vertex of H v and be the graph obtained by taking one copy of F and k vertex disjoint copies of H v , and then attaching the i-th copy of H v to the vertex v i ,i = 1, ..., k, at the vertex v of H (identify v i with the vertex v of the i-th copy).Then the copies of the graph H v that are attached the vertices v i , i = 1, ..., k, are referred to as pockets , and G is described as a graph with k pockets.
Definition 2 [12] Let F and H uv be two graphs of orders n and m , respectively, where n ≥ 2, m ≥ 3, E k = {e 1 , e 2 , • • • e k } is a subset of the edge set of F and H uv has a specified edge uv such that ] be the graph obtained by taking one copy of F and k vertex disjoint copies of H uv , and then pasting the edge uv in the i-th copy of H uv with the edge e i ∈ E k , where i = 1, ..., k.Then the copies of the graph H uv that are pasted to the edges e i , i = 1, ..., k, are called as edge-pockets , and G is described as a graph with k edge pockets.
Note that if a copy of H v is attached to every vertex of F, each at the vertex Bu et al. investigated resistance distance in subdivision-vertex join and subdivision-edge join of graphs [14].Liu et al. [15] gave the resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs.Motivated by the results, in this paper we obtain formulas for resistance distances and Kirchhoff index ] in terms of the resistance distance and Kirchhoff index of F, H v and F, H uv .

Preliminaries
The {1}-inverse of M is a matrix X such that MXM = M.If M is singular, then it has infinite {1}-inverse [17].For a square matrix M, the group inverse of M, denoted by M # , is the unique matrix X such that MXM = M, XMX = X and MX = XM.It is known that M # exists if and only if rank(M) = rank(M 2 ) ( [17], [18]).If M is real symmetric, then M # exists and M # is a symmetric {1}-inverse of M. Actually, M # is equal to the Moore-Penrose inverse of M since M is symmetric [18].
It is known that resistance distances in a connected graph G can be obtained from any {1}inverse of G ([6], [16]).We use M (1) to denote any {1}-inverse of a matrix M, and let (M) uv denote the (u, v)entry of M.
Lemma 1.1 ([6], [18]) Let G be a connected graph.Then Let 1 n denote the column vector of dimension n with all the entries equal one.We will often use 1 to denote all-ones column vector if the dimension can be read from the context.20]) Let G be a connected graph on n vertices.Then be the Laplacian matrix of a connected graph.If D is nonsingular, then

The resistance distance and Kirchhoff index of
In this section, we focus on determing the resistance distance and Kirchhoff index of G[F, V k , H v ] in terms of the resistance distance and Kirchhoff index of F, H v .
Theorem 2.1 Let F, H v be graphs of orders n and m, respectively, where m ≥ 2, v be a specified vertex of H v and (i) For any i, j ∈ V(F), we have (ii) For any i, j ∈ V(H), we have . .
By Lemma 1.5, we have According to Lemma 1.5, we calculate −H # BD −1 and −D −1 B T H # .

−H
We are ready to compute the Lemma 1.5, the following matrix For any i, j ∈ V(F), by Lemma 1.1 and the Equation (1), we have as stated in (i).
For any i, j ∈ V(H), by Lemma 1.1 and the Equation (1), we have as stated in (ii).
For any i ∈ V(F), j ∈ V(H), by Lemma 1.1 and the Equation (1), we have as stated in (iii).By Lemma 1.4, we have Note that the eigenvalues of (L(H) Note that the eigenvalues of (L(H) Let P = (L(H) + I m−1 ) ⊗ I k , then Plugging (2) and (3 , we obtain the required result in (iv).

Resistance distance and Kirchhoff index of G[F, E k , H uv ]
In this section, we focus on determing the resistance distance and Kirchhoff index of G[F, E k , H uv ] in terms of the resistance distance and Kirchhoff index of F, H uv .
Theorem 3.1 Let F and H uv be two graphs of orders n and m , respectively, where n ≥ 2, m ≥ 3, have the resistance distance and Kirchhoff index as follows: (i) For any i, j ∈ V(F), we have (ii) For any i, j ∈ V(H), we have (iii) For any i ∈ V(G), j ∈ V(H), we have Proof Let F S be an r-regular subgraph of F on the first p vertices, then the Laplacian matrix of G = G[F, E k , H uv ] can be written as By Lemma 1.5, we have

Preprints
We are ready to compute the For any i, j ∈ V(F), by Lemma 1.1 and the Equation (4), we have as stated in (i).
For any i, j ∈ V(H), by Lemma 1.1 and the Equation (4), we have as stated in (ii).
For any i ∈ V(F), j ∈ (H), we have

(www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 October 2018 doi:10.20944/preprints201810.0241.v1 Proof Since
v is of degree m − 1, H v can be written as H v = {v} ∨ H, where H is the graph obtained from H v , after deleting the vertex v and the edges incident to it, the Laplacian matrix of G