Bounds for the smallest positive eigenvalue of unicyclic graphs with diameter at most 4

Let G be a simple graph on n vertices and $\tau \left(G\right)$ denote the smallest positive eigenvalue of its adjacency matrix $A\left(G\right)$. In [S. Rani and S. Barik, Upper bounds on the smallest positive eigenvalues of trees, Ann. Comb. 27(1) (2023) 19–29], the authors characterized the trees with small diameters having the maximum and minimum τ, respectively. In this article, we extend their work to the unicyclic graphs. We provide bounds for the smallest positive eigenvalue and obtain the graphs with the maximum and minimum τ among all the unicyclic graphs on n vertices having diameters 2 and 3, respectively. Furthermore, we characterize the graphs with the maximum τ among all the unicyclic graphs on n vertices having diameter 4. Finally, we characterize all the unicyclic graphs on n vertices with diameter at most 4 whose smallest positive eigenvalue is equal to $\frac{\sqrt{5}-1}{2}$, the reciprocal of the golden ratio.