On the distance spectrum of minimal cages and associated distance biregular graphs

TL;DR

This paper derives formulas for the distance spectral radius and eigenvalues of minimal cages and their subdivisions, revealing spectral properties and partially addressing an open problem in graph theory.

Contribution

It provides explicit formulas for the distance spectral radius and eigenvalues of minimal cages and their subdivisions, advancing understanding of their spectral characteristics.

Findings

Distance spectral radius expressed in terms of k and g

Minimal cages have d+1 distinct distance eigenvalues

Full distance spectrum determined for subdivisions of some minimal cages

Abstract

A $(k,g)$-cage is a $k$-regular simple graph of girth $g$ with minimum possible number of vertices. In this paper, $(k,g)$-cages which are Moore graphs are referred as minimal $(k,g)$-cages. A simple connected graph is called distance regular(DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular(DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal $(k,g)$-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal $(k,g)$-cages we give a formula for distance spectral radius in terms of $k$ and $g$, and also determine polynomials of degree $[\frac{g}{2}]$, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal $(k,g)$-cage of diameter $d$ has $d+1$ distinct distance eigenvalues, and this partially answers a problem posed in [5]. We prove that every DBR graph is a $2$-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal $(k,g)$-cages. Finally we determine the full distance spectrum of subdivision of some minimal $(k,g)$-cages.