Kronecker Product Of Graphs Research Articles

On Star Coloring of Kronecker Product of Graphs

On Star Coloring of Kronecker Product of Graphs

The Restricted Edge-Connectivity of Kronecker Product Graphs

The restricted edge-connectivity of a connected graph [Formula: see text], denoted by [Formula: see text], if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, it is proved that [Formula: see text] for any graph [Formula: see text] and a complete graph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is minimum edge-degree of [Formula: see text], and a sufficient condition such that [Formula: see text] is [Formula: see text]-optimal is acquired.

SUPER EDGE CONNECTIVITY OF KRONECKER PRODUCTS OF GRAPHS

A graph G is said to be super edge connected (in short super – λ) if every minimum edge cut isolates a vertex of G. The Kronecker product of graphs G and H is the graph with vertex set V(G × H) = V(G) × V(H), where two vertices (u1, v1) and (u2, v2) are adjacent in G × H if u1u2 ∈ E(G) and v1v2 ∈ E(H). Let G be a connected graph, and let δ(G) and λ(G) be the minimum degree and the edge-connectivity of G, respectively. In this paper we prove that G × Kn is super-λ for n ≥ 3, if λ(G) = δ(G) and G ≇ K2. Furthermore, we show that K2 × Kn is super-λ when n ≥ 4. Similar results for G × Tn are also obtained, where Tn is the graph obtained from Kn by adding a loop to every vertex of Kn.

Some remarks on the Kronecker product of graphs

This short note is concerned with the Kronecker product of graphs; we give some properties linked to graph minors, planarity, cut vertex and cut edge.

The Kronecker product of graphs

Introduction. This note considers a product derived from the Kronecker product of matrices. Some indication of the geometrical nature of this product is given and a theorem stating necessary and sufficient conditions for a product to be connected is proved. The matrix analogue of the above result is also stated. I. A convenient representation for a finite undirected [1] G is an adjacency matrix. If the vertex set of G is { Pi }, i = 1, , then an adjacency matrix of G is an nXn matrix (aij) with ai1 equal to the number of lines (paths of length one) joining pi to pj. A given is then represented by an equivalence class of matrices A= { PiAPi-l for all permutation matrices Pi of order equal to the order of A }. Each element of A: corresponds to a different ordering of the vertices of G. It is clear that for each such class of adjacency matrices there corresponds a unique class of isomorphic graphs. From this point on graph will mean a finite undirected with no loops. Such a has an adjacency matrix (ai0) whose entries are non-negative integers such that aij=aji and ai=0. We also use the following notation: o(G) is the number of vertices of G and is called the order of G, pLp7, is a chain in G from vertex Pi to vertex Pk, and n(pl->pk) is the number of lines (not necessarily distinct) in PlPk. If A and B are two adjacency matrices and A o B is some matrix product which is also an adjacency matrix then this matrix operation