The generalized hierarchical product of graphs

Abstract

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

On Weighted PI Index of Graphs

“…Barriere et al [4,5] defined a new product of graphs, namely, the generalized hierarchical product, as follows: Let G and H be two graphs with a nonempty vertex subset U ⊆ V (G). Then the generalized hierarchical product, denoted by G(U ) H, is the graph with vertex set V (G) × V (H) and two vertices (g, h) and (g , h ) are adjacent if and only if g = g ∈ U and hh ∈ E(H) or, gg ∈ E(G) and h = h .…”

Local metric dimension of graphs: Generalized hierarchical products and some applications

“…(Note that this vertex from U could be one of u and v.) With d G(U ) (u, v) we denote the length of a shortest u, v-walk through U. The following fundamental observation from [3] will be used throughout the paper, mostly without explicitly mentioning it.…”

Spectra, Hitting Times and Resistance Distances ofq- Subdivision Graphs

“…designing models for complex networks. Frequently used graph operations and products include edge iteration [8,9], planar triangulation [10,11,12], Kronecker product [13,14,15], hierarchical product [16,17,18], and corona product [19,20,21].…”