On the monotonicity of topological indices and the connectivity of a graph

On extremal bipartite graphs with a given connectivity

“…Gutman and Zhang [15] determined the graph with minimum Wiener index among all graphs of order n and vertex (edge) connectivity k. The extremal values of Zagreb and hyper-Wiener indices with a given connectivity obtained by Behtoei et al [1]. Li and Zhou [19] studied the extremal properties of the first and second Zagreb index when connectivity is at most k. More results for graphs with a given connectivity can be found in [27,29,31] and the references cited therein. In 2012, Nath and Paul [22] determined the minimum distance spectral radius among all connected bipartite graphs of order n with a given matching number and a vertex connectivity.…”

“…After that, these results for bipartite graphs haven been extended to the distance Laplacian spectral radius by Liu et al [23], to the Wiener index by Li and Song [18] and the Estrada index by Huang et al [16]. On the basis of the monotonicity, unified approaches for various topological indices of graphs (or bipartite graphs) in terms of some kinds of graph parameters were proposed in [4,5,29].…”

Bounds on some monotonic topological indices of bipartite graphs with a given number of cut edges

“…It is well known that many important topological indices have the monotonicity [1,2,11], i.e., they decrease (or increase, respectively) with addition of edges, including the Wiener index, the Kirchhoff index, the Hosoya index, the Estrada index, the Zagreb index etc.…”

“…In fact, all these indices in these literatures above are monotonic. In light of the monotonicity, the present authors [11] determined the extremal values of some monotonic topological indices in terms of the number of cut vertices, or number of cut edges, or the vertex connectivity, or the edge connectivity of a graph. In [2], the authors determined the extremal values of some monotonic topological indices in graphs with a given vertex bipartiteness.…”

Extremal Results and Bounds for Atom-Bond Sum-Connectivity Index

“…. , e r } is either a complete bipartite graph or K 1 .Theorem 43 [97]. If G is a graph attaining the maximum ABS index among all connected n-order graphs with r cut vertices, then every block of G is complete and every cut vertex of G is contained in exactly two blocks.Problem 5.…”