On Extended Adjacency Matrices Associated with Eccentricity-Based Topological Indices

Several investigations of the extended adjacency matrices associated with various degree-based topological indices have been undertaken in recent years, leading to sharper theorems and conclusions. Motivated by this, we explore eccentricity-based topological indices and the extended adjacency matrices associated with them. Suppose F is a symmetric function associated with an eccentricity-based topological index, then a generalised extended adjacency matrix A^e_F(G) has (i,j)^th- entry as the value of F at the eccentricities of the corresponding vertices v_i,v_j if they are adjacent, else it is 0. The expressions and bounds for parameters like the trace, determinant, and eigenvalues associated with this matrix are derived in this article. The expression for the determinant indicates that it depends on the number of elementary spanning subgraphs present in the graph.