Spectral sufficient conditions of the existence of the longest path in the graph and its traceability

Abstract

It is known that the existence of many classical combinatorial structures in a graph, such as perfect matchings, Hamiltonian cycles, effective dominating sets, etc., can be characterized using (κ,τ)-regular sets, whose determination is equivalent to the determination of these classical combinatorial structures. On the other hand, the determination of (κ,τ)regular sets is closely related to the properties of the main spectrum of a graph. We use the previously obtained generalized properties of (κ,τ)-regular sets of graphs to develop a recognition algorithm of the traceability of a graph. We also obtained new sufficient conditions for the existence of a longest simple path in a graph in terms of the spectral radius of the adjacency matrix and the signless Laplacian of the graph.