Alireza Abdollahi - University of Isfahan

‎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 ۹ 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_۱,\dots,d_n\} is the set of vertex degrees of G‎, ‎then‎ ‎\gcd(۲m,d^۲) divides the determinant of the adjacency matrix of G‎, ‎where d=\gcd(d_۱,\dots,d_n)‎. ‎Possible determinants of adjacency matrices of graphs with exactly two cycles are obtained‎.

Determinant, adjacency matrices of graphs, maximum determinant