Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

Abstract

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let U n,g be the lollipop graph obtained by appending a pendent vertex of a path on n − g (n > g) vertices to a vertex of a cycle on g ⩾ 3 vertices. We prove that the graph U n,g uniquely minimizes the Laplacian spectral radius for n ⩾ 2g − 1 when g is even and for n ⩾ 3g − 1 when g is odd.