On the spectral radius of minimally 2-(edge)-connected graphs with given size

Abstract

A graph is minimally $k$-connected ($k$-edge-connected) if it is $k$-connected ($k$-edge-connected) and deleting arbitrary chosen edge always leaves a graph which is not $k$-connected ($k$-edge-connected). A classic result of minimally $k$-connected graph is given by Mader who determined the extremal size of a minimally $k$-connected graph of high order in 1937. Naturally, for a fixed size of a minimally $k$-(edge)-connected graphs, what is the extremal spectral radius? In this paper, we determine the maximum spectral radius for the minimally $2$-connected ($2$-edge-connected) graphs of given size, moreover the corresponding extremal graphs are also determined.