ON INVERTIBLE NON-BIPARTITE UNICYCLIC GRAPHS WITH A UNIQUE PERFECT MATCHING AND THEIR SMALLEST POSITIVE EIGENVALUES

Let G be a simple connected graph with the adjacency matrix A(G). By the smallest positive eigenvalue of G, we mean the smallest positive eigenvalue of A(G) and denote it by tau(G). Recently, the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching has been studied, and the extremal graphs having the minimum and the maximum tau values have been characterized. We consider the same problem for non-bipartite case. A graph G is said to be positively invertible (respectively, negatively invertible) if there exists a signature matrix S such that SA(G)S-1 is nonnegative (respectively, nonpositive). In [S. Akbari and S.J. Kirkland. On unimodular graphs. Linear Algebra Appl., 421:3-15, 2007], the authors characterized all the bipartite unicyclic graphs with a unique perfect matching that are positively invertible. In this article, we characterize all the non-bipartite unicyclic graphs with a unique perfect matching that are positively invertible and negatively invertible, respectively. As an application, we obtain the unique graph with the minimum tau among all the non-bipartite unicyclic graphs on n vertices with a unique perfect matching. Except for a specific class, we characterize all other non-bipartite unicyclic graphs G with a unique perfect matching such that tau(G) < 1/2 . Further, we show that if G is a non-bipartite unicyclic graph with a unique perfect matching, then tau(G) <= root 5-1 /2 . The extremal graphs with tau = root 5-1 /2 have been obtained. Finally, we obtain the graphs with the maximum tau among all the non-bipartite unicyclic graphs on n vertices with a unique perfect matching.