A HYBRID TOPOLOGICAL INDEX BASED ON LAPLACIAN AND DISTANCE SPECTRA

Abstract

In graph theory, the topological indices translate complex combinatorial properties such as connectivity, distances, branching, and symmetry into single quantitative measures. They play a crucial role in comparing graphs, studying extremal structures, analyzing chemical molecules, and developing models in network science. By summarizing key aspects of a graph’s topology, the indices provide a powerful tool for both theoretical investigations and real-world applications.

In this paper, the authors introduce a new topological index for finite simple connected graphs which is called as Distance-Spectral Mixed Index (DSMI), that combines two different spectra, the Laplacian and distance spectra. Let G be a connected graph of order  with Laplacian eigenvalues  and distance-matrix eigenvalues  The is defined as  The authors discussed the invariance and nonnegativity, upper and lower bounds in terms of classical invariants like order, diameter, algebraic connectivity, established results on monotonicity under elementary graph operations such as addition of edge and vertex, subdivision of edge and the exact values for complete graphs, complete bipartite graphs, paths, and stars are obtained.