Abstract
Let be a simple graph with vertex set and edge set . The signature of is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix . In [22, Linear Algebra Appl. 2013;438:331–341], it was conjectured that , where denotes the number of cycles in of length , and denotes the number of cycles in of length . The authors of this paper have established the conjecture for trees, unicyclic graphs and bicyclic graphs. We prove the conjecture for -cyclic graphs of -type, that is, the graphs in which and in which no two cycles have a common vertex.
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