On the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching

Abstract

Let G be a simple graph with the adjacency matrix A(G). Let τ(G) denote the smallest positive eigenvalue of A(G). In 1985, Godsil proved that in the class of all nonsingular trees on n=2m vertices, the path on 2m vertices has the minimum τ value. We consider the same problem for unicyclic graphs. Let Un be the class of connected bipartite unicyclic graphs on n=2m vertices, with a unique perfect matching. Let us take the path P2m=[1,2,…,m,m+1,…,2m]. If m is even, let Ue be the graph obtained from P2m by adding an edge between the vertices m−2 and m+3. If m is odd, let Uo be the graph obtained from the path P2m by adding an edge between the vertices m−3 and m+2. We show that the unique graph Ue (resp. Uo) has the minimum τ value among all the graphs in U2m when m is even (resp. when m is odd). In 1990, Pavlíková and Krč-Jediný proved that among all nonsingular trees on n=2m vertices the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. Thus, it can be observed that there are only three trees with the smallest positive eigenvalue greater than 12. In this article, we characterize all bipartite unicyclic graphs G with a unique perfect matching such that τ(G)<12 (ultimately, we obtain only two bipartite unicyclic graphs with a unique perfect matching such that τ(G)>12).