Maximal Spectral Radius of Minimally k -(Edge)-Connected Graphs

Abstract

Minimally $k$-connected graphs are the main focus of both structural and extremal graph theory. Perhaps the most heavily investigated parameter of this graph family is the number $\mid V_k\mid$ of vertices of degree $k$. Mader proved a tight lower bound for $\mid V_k\mid$, independent of $k$, and the order $n$. In 1981, inspired by matroids, Oxley discovered that in many cases, a considerably better bound can be given by using the size $m$ as a parameter. Along this line, Schmidt [Tight bounds for the vertices of degree $k$ in minimally $k$-connected graphs, J. Graph Theory 88 (2018) 146–153] showed that $\mid V_k\mid \ge \max \left\{\left(k+1\right)n-2m,\lceil \left(m-n+k\right)∕\left(k-1\right)\rceil \right\}$, and this bound is best possible. Another interesting problem was posed for connected graphs with fixed size: what is the maximal spectral radius of a minimally $k$-(edge)-connected graph on $m$ edges? This contribution can be traced back to Brualdi and Hoffman, who also conjectured that $G_m$ is the extremal graph among all connected graphs with $m$ edges, where $G_m$ is obtained from the complete graph $K_s$ by adding a new vertex of degree $m-\left(\frac{s}{2}\right)$. This conjecture was completely solved by Rowlinson in 1988 using double eigenvector transformations. Recently, the case for $k=2$ was answered by Lou, Gao and Huang (2023). In this paper, we solve the problem completely, and further, for each $k\ge 2$ and $m=\Omega \left(k^6\right)$, the unique extremal graph is determined.