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 $SA(G)^{-1}S$ 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 $\tau(G)<\frac{1}{2}$. Further, we show that if $G$ is a non-bipartite unicyclic graph with a unique perfect matching, then $\tau(G)\leq\frac{\sqrt{5}-1}{2}$. The extremal graphs with $\tau=\frac{\sqrt{5}-1}{2}$ 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.