2023/01/04 by Iswar Mahato, M. Rajesh Kannan, Mahato, Iswar +1
Chemistry · Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Synthesis and Properties of Aromatic Compounds
paper · pdf · doi:10.48550/arxiv.2301.01708
openalex publication_date 2023/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The eccentricity matrix of a connected graph G, denoted by E(G), is obtained from the distance matrix of G by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The E-eigenvalues of G are the eigenvalues of E(G), in which the largest one is the E-spectral radius of G. The E-energy of G is the sum of the absolute values of all E-eigenvalues of G. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) E-spectral radius among the complements of trees. Then, we prove that the E-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least E-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest E-eigenvalue and the E-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum type lower bounds for the second largest E-eigenvalue and E-energy of a tree and its complement.