Sufficient conditions for graphs to be $k$-connected, maximally connected and super-connected

TL;DR

This paper establishes conditions based on edges count and spectral radius under which graphs are guaranteed to be k-connected, maximally connected, or super-connected, enhancing understanding of graph connectivity properties.

Contribution

It provides new sufficient conditions involving edge number and spectral radius for graphs to be k-connected, maximally connected, or super-connected, including for triangle-free graphs.

Findings

Graphs with enough edges are k-connected.

Large spectral radius implies high connectivity.

Conditions apply to triangle-free graphs.

Abstract

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is $k$-connected, maximally connected or super-connected if the number of edges or the spectral radius is large enough.