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Determinants of adjacency matrices of graphs. (English) Zbl 1272.05107
Summary: 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 [Can. J. Math. 30, 756–762 (1978; Zbl 0388.15012)], 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.

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A99 Basic linear algebra
15A15 Determinants, permanents, traces, other special matrix functions
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