Determinants of adjacency matrices of graphs

Document Type : Research Paper


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 $9$ 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.


Main Subjects

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Volume 1, Issue 4 - Serial Number 4
December 2012
Pages 9-16
  • Receive Date: 16 June 2012
  • Revise Date: 24 November 2012
  • Accept Date: 28 November 2012
  • Published Online: 01 December 2012