Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 8 2 2024 10 22 The Maximum Edge Eccentricity Energy of a Graph 182 190 438 10.22128/gadm.2024.863.1119 EN Akram Sadat Banihashemi Dehkordi Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195--363, Semnan, Iran Saeed Mohammadian Semnani Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box: 35195--363, Semnan, Iran 0000-0001-6755-4911 Journal Article 2024 08 19 This paper presents a new concept in graph theory‎, ‎focusing on a connected graph's edge eccentricity‎. ‎We define a new matrix‎, ‎the maximum edge eccentricity matrix $M_{e_{e}}(\Upsilon)$‎, ‎which represents the maximum edge distance between all pairs of edges in the graph‎. ‎This matrix is derived from the graph's structure and the eccentricity values of its edges‎. ‎Our work explores the characteristics of this matrix‎, ‎including the determination of specific coefficients within its characteristic polynomial‎, ‎denoted as $P(\Upsilon,\nu)$‎. ‎Furthermore‎, ‎we introduce the concept of maximum edge eccentricity energy $M_{e_{e}}(\Upsilon)$ for connected graphs and provide calculations for well-known graphs‎. ‎We establish upper and lower bounds for $E_{M_{e_{e}}}(\Upsilon)$ and prove that if the maximum edge eccentricity energy of a graph is rational‎, ‎it must be an even number. https://ansne.du.ac.ir/article_438_993af8aea015cb7aa068139e9f259e5e.pdf