Extremal Laplacian Spectral Radius in Cactus Graphs

A cactus is a connected graph in which any two cycles have at most one common vertex, making it a natural extension of trees and unicyclic graphs. In this paper, we investigate the extremal behavior of the Laplacian spectral radius among all cactus graphs with n vertices. It is known that the path graph achieves the minimum Laplacian spectral radius among all n-vertex cacti, we show that certain friendship-like cacti maximize it. We identify the cycle graph and the tadpole graph as attaining the smallest and second smallest values, respectively, among non-tree cacti. This article extend classical extremal results to the spectral characterization of cactus graphs.