%0 Journal Article
%T Determinants of adjacency matrices of graphs
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Abdollahi, Alireza
%D 2012
%\ 12/01/2012
%V 1
%N 4
%P 9-16
%! Determinants of adjacency matrices of graphs
%K Determinant
%K adjacency matrices of graphs
%K maximum determinant
%R 10.22108/toc.2012.2041
%X 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.
%U https://toc.ui.ac.ir/article_2041_b9579dd3348af7ab9be7b60997202498.pdf