TY - JOUR
ID - 2041
TI - Determinants of adjacency matrices of graphs
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Abdollahi, Alireza
AD - University of Isfahan
Y1 - 2012
PY - 2012
VL - 1
IS - 4
SP - 9
EP - 16
KW - Determinant
KW - adjacency matrices of graphs
KW - maximum determinant
DO - 10.22108/toc.2012.2041
N2 - 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.
UR - https://toc.ui.ac.ir/article_2041.html
L1 - https://toc.ui.ac.ir/article_2041_b9579dd3348af7ab9be7b60997202498.pdf
ER -