Determinants of adjacency matrices of graphs

• Main Author: Alireza Abdollahi
• Format: Article
• Language: English
• Published: University of Isfahan 2012-12-01
• Series: Transactions on Combinatorics
• Subjects: Determinant; adjacency matrices of graphs; maximum determinant
• Online Access: http://www.combinatorics.ir/?_action=showPDF&article=2041&_ob=dfea7756845067696c4a750520b02fe7&fileName=full_text.pdf.

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 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.