Wiener-type invariants of pancyclicity for t-tough graphs

A connected graph G is t-tough if t <middle dot> c(G-S) <= |S| for every vertex cut S of G, where c(G-S) is the number of components of G-S. A graph G is called pancyclic if it has cycles of all lengths from 3 to n. A Wiener-type invariant of a connected graph G is defined as Wf = Sigma u,vEV(G)f(dG(u, v)), where f(x) is a nonnegative function on the distance dG(u, v). In this paper, we present the best possible Wiener-type conditions to guarantee a ttough graph to be pancyclic in the case when t E {1, 2, 3}. Furthermore, we determine sufficient conditions on the distance and distance signless Laplacian spectral radius for a graph to be pancyclic. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.