Determinants of adjacency matrices of graphs

Document Type: Research Paper

Author

University of Isfahan

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.

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