Abstract
Let G[F,Vk,Hv] 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, Hv is a simple graph of order m≥2, and 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, Huv is a simple graph of order m≥3, and 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.
Highlights
There are many theories and methods for studying complex networks
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 [12] and [3, 8], respectively
Liu et al [14, 15] gave the resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs
Summary
There are many theories and methods for studying complex networks. Scholars describe and display complex networks from various angles and strive to discover new phenomena and new features of complex networks. Resistance distance and Kirchhoff, the advantage of the index is that it can better reflect the integrity and connectivity of the network. With these two Mathematical Problems in Engineering quantities, we can optimize the network. 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 [12] and [3, 8], respectively. Liu et al [14, 15] gave the resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs.
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