On the smallest positive eigenvalue of caterpillar unicyclic graphs

Abstract

Let $G$ be a simple graph with the adjacency matrix $A\left(G\right)$. By the smallest positive eigenvalue of $G$, we mean the smallest positive eigenvalue of $A\left(G\right)$ and denote it by $\tau \left(G\right)$. For $k\ge 3$, let $C_k$ be the cycle graph with vertices $1,2,\dots ,k$ and $n_1,n_2,\dots ,n_k$ be $k$ nonnegative integers. A caterpillar unicyclic graph $C_k(n_1,n_2,\dots ,n_k)$ is a graph obtained from $C_k$ by adding $n_i$ pendant vertices to the vertex $i$ of $C_k$, for $i=1,2,\dots ,k$. Let $C_n$ be the class of all caterpillar unicyclic graphs on $n$ vertices, where each $n_i$ is positive. In this article, we obtain the graphs with the maximum $\tau$ among all the graphs in $C_n$. Furthermore, we characterize the graphs $G$ in $C_n$ such that $\tau \left(G\right)=\sqrt{2}-1$ and $\tau \left(G\right)>\sqrt{2}-1$, respectively. As a consequence, we obtain the graphs with the minimum $\tau$ among all the graphs in $C_n$. Let ${\rm Y}$ be the set of all smallest positive eigenvalues of the caterpillar unicyclic graphs in $C_n$. We show that the largest limit point of ${\rm Y}$ is not finite and the smallest limit point of ${\rm Y}$ is $\sqrt{2}-1$.