An upper bound for the Laplacian index of a signed graph

Abstract

We prove that µ1 ≤ max {√ 2(di2 + dimi - 2Ti+): 1 ≤ i ≤ n}, where µ1 is the Laplacian index of a signed graph Ġ with n vertices and, for a vertex i, the symbols di, mi and T+i denote its degree, average 2-degree and the number of positive triangles containing i, respectively. We also show that equality holds if and only if Ġ is switching equivalent to a regular signed graph with all edges being negative. Apart from this result, we derive some other upper bounds for µ1, make some comparisons and conclude by finding a lower bound for the same eigenvalue.