Determinants of adjacency matrices of graphs

Abstract

‎We study the set of all determinants of adjacency matrices of graphs with a given number of vertices‎. ‎Using Brendan McKay's data base of small graphs‎, ‎determinants of graphs with at most $9$ vertices are computed so‎ ‎that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table‎. ‎Using an idea of M‎. ‎Newman‎, ‎it is proved that if $G$ is a graph with $n$ vertices‎, ‎$m$ edges and ${d_1,dots,d_n}$ is the set of vertex degrees of $G$‎, ‎then‎ ‎$gcd(2m,d^2)$ divides the determinant of the adjacency matrix of $G$‎, ‎where $d=gcd(d_1,dots,d_n)$‎. ‎Possible determinants of adjacency matrices of graphs with exactly two cycles are obtained‎.