Permutational Powers of a Graph

Matteo Cavaleri,Alfredo Donno,Daniele D'Angeli

doi:10.37236/8337

Abstract

This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction recovers the classical zig-zag product of graphs when the permutation is an involution, and it is in fact more general. We start by discussing necessary and sufficient conditions on the permutation and on the adjacency matrix of a graph to guarantee their composition to represent an adjacency matrix of a graph, then we focus our attention on the cases in which the permutational power does not reduce to a zig-zag product. We show that the cases of interest are those in which the adjacency matrix is singular. This leads us to frame our problem in the context of equitable partitions, obtained by identifying vertices having the same neighborhood. The families of cyclic and complete bipartite graphs are treated in details.

Highlights

Graphs are among the most popular and useful tools used in Mathematics to model aspects of real life
Graphs are studied via their adjacency matrix, and it is an interesting task to investigate the relationship between the geometrical properties of a graph and the algebraic properties of the corresponding adjacency matrix
It is clear that, whenever we have AHP AH symmetric we can ask if the same graph can be obtained via the “classical” zig-zag product

Summary

Introduction

Graphs are among the most popular and useful tools used in Mathematics to model aspects of real life. It is clear that, whenever we have AHP AH symmetric (and so it can interpreted as the adjacency matrix of an undirected graph) we can ask if the same graph can be obtained via the “classical” zig-zag product (by using a symmetric P ). A permutational power of H with respect to p can be obtained by a classical zig-zag product if there exists an involution q sharing with p the same projection on the range of the adjacency matrix of H. The graph obtained collapsing the equivalent vertices together (in a precise way), can be singular or not In the latter case every permutational power of H can be obtained via the classical zig-zag product (Theorem 27): the easiest examples are the complete bipartite graphs. The cases in which H is a cycle or a complete bipartite graph are studied in details (see Sections 3.2 and 3.3, and Section 4.1)

Preliminaries and motivations

An algebraic interpretation

Cyclic graphs

Equitable partitions

The complete bipartite graph