Applications of E‐graphs in network design

Abstract

In this paper, we introduce a construction that produces graphs, called €-graphs, by replacing the edges in a core graph with a copy of a given graph. These graphs are generalizations of several graphs that have recently been presented as models for network designs, including the G-network and its extensions. We discuss several invariant properties of these graphs with emphasis on those of interest in network design, such as number of edges, diameter, and domination number. 0 7993 by John Wiley 8 Sons, Inc.

Edge corona product as an approach to modeling complex simplical networks

“…The network family proposed and studied here is constructed based on the edge corona product of graphs defined as follows [53], [54], [55], which is a variant of the corona product first introduced by Frucht and Harary [56] of two graphs. Let G 1 and G 2 be two graphs with disjoint vertex sets, with the former G 1 having n 1 vertices and m 1 edges.…”

“…In this paper, by literately applying edge corona product [53] first proposed by Haynes and Lawson [54], [55] to complete graphs or q-cliques K q with q ≥ 1, we propose a mathematically tractable model for complex networks with various cycles at different scales. Since the resultant networks are composed of cliques of different sizes, we call these networks as simplical networks.…”

Computing Certain Degree Based Topological Indices and Coindices of E-graphs

“…Designate two nodes ‫ݔ‬ ଵ and ‫ݔ‬ ଶ , ‫ݔ‬ ଵ ≠ ‫ݔ‬ ଶ in H as e-nodes such that there is an automorphism ߪ: ‫)ܪ(ܸ‬ → ‫)ܪ(ܸ‬ with the property ‫ݔ(ߪ‬ ଵ ) = ‫ݔ‬ ଶ ; ‫ݔ(ߪ‬ ଶ ) = ‫ݔ‬ ଵ . Then the symmetric edge replacement of ‫ܩ‬ by ‫ܪ‬ written as ‫,ܪ|ܩ‬ is the ‫-ܧ‬graph got by replacing every edge ‫ݒݑ‬ of ‫ܩ‬ with a copy of ‫ܪ‬ identifying ‫ݑ‬ and ‫ݒ‬ with ‫ݔ‬ ଵ and ‫ݔ‬ ଶ respectively [7]. |ܹ ହ where two nonadjacent vertices of degree three in nodes is called as the extended G-network [14].…”

“…Figure1: E-graph ‫.ܪ|ܩ‬ The ‫-ܧ‬graph ‫ܭ‬ ‫ܥ|‬ ସ where the e-nodes are two nonadjacent vertices of ‫ܥ‬ ସ , is called as the ‫-ܩ‬network[7].…”

$T$ and $St$-Coloring of Corona Product and Edge Corona Product of Graphs

“…In recent years, the Cartesian product, [13] hierarchical product, Kronecker product, [7], and others have been employed to emulate complex networks. The Corona Product [5] and Edge Corona Product (or Edge Corona network) [8] are also helpful for creating a big graph from smaller ones. They are new in mathematics.…”