On So's conjecture for integral circulant graphs

J.W Sander,T Sander

doi:10.2298/aadm150226009s

Abstract

Each integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z=nZ and edge set {(a,b) : a, b ? Z=nZ, gcd(a - b,n) ? D}. According to a conjecture of So two integral circulant graphs are isomorphic if and only if they are isospectral, i.e. they have the same eigenvalues (counted with multiplicities). We prove a weaker form of this conjecture, namely, that two integral circulant graphs with multiplicative divisor sets are isomorphic if and only if their spectral vectors coincide.

Highlights

AND RESULTSThe vivid question “Can one hear the shape of a drum?”, posed by Kac [11] in 1966, has become a synonym for the considerably older and much more general problem to decide whether a Riemannian manifold is determined by its spectrum
Originating from a problem in chemistry [8], it has been asked in general since the mid-twentieth century which graphs are determined by their spectrum, and some answers were given for different types of graphs. Our paper addresses this problem with regard to the class of integral circulant graphs and partially proves a conjecture of So [19]
It is easy to find non-isomorphic integral circulant graphs of the same order with different divisor sets having the same spectrum if we neglect the multiplicities of the eigenvalues

Summary

AND RESULTS

The vivid question “Can one hear the shape of a drum?”, posed by Kac [11] in 1966, has become a synonym for the considerably older and much more general problem to decide whether a Riemannian manifold is determined by its spectrum. 7]), assuming the graphs to have no loops This can be shown quite for instance by verifying that two loopless integral circulant graphs of the same prime power order with different divisor sets have different largest eigenvalues (see Corollary 2.2 below). It is easy to find non-isomorphic integral circulant graphs of the same order with different divisor sets having the same spectrum if we neglect the multiplicities of the eigenvalues. The spectral vector of an integral circulant graph ICG(n, D) with an arbitrary positive integer n and arbitrary divisor set D ⊆ D(n) is defined as λ(n, D) := λ1(n, D), λ2(n, D), . In order to give an impression of the significance of Theorem 1.2 as compared to the original Conjecture of So we determine the proportion of multiplicative divisor sets among all divisor sets for any given positive integer n. Apart from that Theorem 1.2 is the only non-trivial case where the Conjecture of So could be verified so far, namely for graphs ICG(n, D) with arbitrary n and arbitrarily large divisor sets D

SPECTRAL VECTORS FOR ICGS OF PRIME POWER ORDER

SPECTRAL VECTORS FOR ICGS OF ARBITRARY ORDER