Variable neighbourhood search for connected graphs of fixed order and size with minimal spectral radius

Abstract

In this study we consider connected graphs of fixed order n and size m that minimize the largest eigenvalue of the adjacency matrix, also known as the spectral radius. Such graphs are called minimizers. The motivation for this research lies in the fact that the spectral radius plays a significant role in modelling virus propagation in complex networks, in the sense that a smaller spectral radius ensures a better virus protection in the network modelled by the corresponding graph. We conjecture that vertex degrees of a minimizer are as equal as possible, i.e. belong to {⌊2m/n⌋,⌈2m/n⌉}. The conjecture is confirmed for graphs with at most 10 vertices by the total enumeration, and some classes of graphs are resolved theoretically. In general, exact determining of minimizers is quite difficult and computationally exhaustive, and thus we propose a long-scale variable neighbourhood search as an alternative approach. We employ this heuristic search for selected instances concerning graphs with at most 100 vertices, and as a result we always obtain a graph with required vertex degrees. The search efficiently results in solutions that are (in a precise sense) close to the optimal ones, i.e. to minimizers.