Lower bounds for the least Laplacian eigenvalue of unbalanced blocks

Abstract

We denote by n and μn the number of vertices and the least Laplacian eigenvalue of a signed graph, respectively. A connected unbalanced signed graph without cut-vertices is called an unbalanced block. We prove that [Formula presented] holds for every unbalanced block G˙, where lu denotes the length of the longest negative cycle in G˙. We also prove that [Formula presented] (g(n1,n2,…,nk) being the geometric mean of given arguments) holds for every signed graph G˙ which contains k edge-disjoint spanning subgraphs such that the least Laplacian eigenvalue of the ith of them is not less than the least Laplacian eigenvalue of the negative cycle C˙ni. Using this result, we prove that [Formula presented] holds for every unbalanced block with k edge-disjoint negative Hamiltonian cycles.