On irregularity indices and main eigenvalues of graphs and their applicability

Graph-theoretic irregularity indices have diverse applications in computer science. This paper extends their practical applicability in reticular chemistry. First, we put forward a method of computing various irregularity indices of graphs by means of their main eigenvalues. This presents applications of spectral graph theory in chemistry. We find parametric conditions for which the generalized friendship graphs, the join of two regular graphs, the corona of two regular graphs and the vertex-deleted subgraph of a strongly regular graph have exactly two distinct main eigenvalues. By computing the two main eigenvalues for these classes of graphs, we determine their certain irregularity indices. Our results generalize most of the results of Réti (Appl Math Comput 344–345:107–115, 2019), in which the author studied these irregularity indices for the complete bipartite graphs, friendship graphs and complete split graphs. More importantly, we prove a conjecture proposed in Réti (Appl Math Comput 344–345:107–115, 2019) stating that the complete split-like graphs have exactly two distinct main eigenvalues. At last, we provide the QSPR analysis with regression modeling for irregularity indices with significant predictive potential.