Abstract
Edge-betweenness centrality measure is an important centrality measure that focuses on the role of edges in a graph. The total edge-betweenness centrality measure (EBCM) of a graph is defined as the sum of the edge-betweenness centrality measures of all its edges. In this work, we derive an explicit expression for the total EBCM of the shadow graph of a graph G in terms of the total EBCM of G. Moreover, we establish an interesting result showing that the total EBCM of a graph is equal to its Wiener index. Furthermore, for certain families of graphs of order n, we identify the graphs that attain the minimum and maximum values of the total EBCM. The results provide deeper insights into network connectivity and structural efficiency, offering mathematical tools that can support the development of more resilient, efficient, and sustainable real-world systems.