# Spectral Bounds for the Connectivity of Regular Graphs with Given Order

## TL;DR

This paper establishes new spectral bounds for regular graphs that ensure specific connectivity levels, linking eigenvalues to robustness measures and answering a question posed by Mohar.

## Contribution

The paper introduces two upper bounds for the second-largest eigenvalues of regular graphs with given order, guaranteeing certain connectivity properties, and characterizes cases of equality.

## Key findings

- Derived bounds hold for regular graphs and multigraphs of specified order.
- Bounds are tight for infinite families of graphs.
- Results relate eigenvalues to vertex- and edge-connectivity.

## Abstract

The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, we present two upper bounds for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results answer a question of Mohar.