Proof of a conjecture on the ϵ-spectral radius of trees

The ϵ-spectral radius of a connected graph is the largest eigenvalue of its eccentricity matrix. In this paper, we identify the unique n-vertex tree with diameter 4 and matching number 5 that minimizes the ϵ-spectral radius, and thus resolve a conjecture proposed in [W. Wei, S. Li, L. Zhang, Characterizing the extremal graphs with respect to the eccentricity spectral radius, and beyond, Discrete Math. 345 (2022) 112686]

On the eccentricity energy and eccentricity spectral radius of graphs with odd diameter

The eccentricity matrix of a graph is defined as the matrix obtained from its distance matrix by retaining the largest elements in each row and column, while the rest elements are set to be zero. The eccentricity eigenvalues of a graph are the eigenvalues of its eccentricity matrix, the eccentricity energy of a graph is the sum of the absolute values of its eccentricity eigenvalues, and the eccentricity spectral radius of a graph is its largest eccentricity eigenvalue. Let ${\mathcal{G}}_{n,d}$ be the set of n-vertex connected graphs with odd diameter d, where each graph G in ${\mathcal{G}}_{n,d}$ has a diametrical path whose center edge is a cut edge of G. For any graph G in ${\mathcal{G}}_{n,d}$, we construct a weighted graph Hω such that its adjacency matrix is just the eccentricity matrix of G, where H is the sequential join graph of the complement graphs of four disjoint complete graphs. In terms of the energy and spectral radius of the weighted graphs, we determine the graphs with minimum eccentricity energy, minimum and maximum eccentricity spectral radius in ${\mathcal{G}}_{n,d}$, respectively. As corollaries, we determine the trees with minimum eccentricity energy, minimum and maximum eccentricity spectral radius among all trees with odd diameter, respectively