On the least eccentricity eigenvalue of graphs

Given a connected graph G with vertex set V(G), the distance matrix of G is the matrix D(G) = (dG(u, v))u,vEV(G), and the eccentricity matrix of G is defined as the matrix constructed from the distance matrix of G by keeping for each row and each column the largest entries and setting all other entries to be zero, where dG(u, v) denotes the distance between u and v in G. The eccentricity eigenvalues of G are the eigenvalues of the eccentricity matrix. By interlacing theorem, the least eccentricity eigenvalue of a graph with diameter d is at most -d. We show that this bound is achieved for d > 3 if and only if the graph is an antipodal graph with equal diameter and radius, which solves an open problem proposed in Wang et al. (2020). Then we determine all n-vertex unicyclic graphs and bicyclic graphs that maximize the least eccentricity eigenvalue, respectively.(c) 2023 Elsevier B.V. All rights reserved.