The Maximum Edge Eccentricity Energy of a Graph

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 mee (υ), 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(υ, ν). furthermore, we introduce the concept of maximum edge eccentricity energy mee (υ) for connected graphs and provide calculations for well-known graphs. we establish upper and lower bounds for emee (υ) and prove that if the maximum edge eccentricity energy of a graph is rational, it must be an even number.