On Variable Sum Exdeg Indices of Quasi-Tree Graphs and Unicyclic Graphs

Abstract

In this work, by using the properties of the variable sum exdeg indices and analyzing the structure of the quasi-tree graphs and unicyclic graphs, the minimum and maximum variable sum exdeg indices of quasi-tree graphs and quasi-tree graphs with perfect matchings were presented; the minimum and maximum variable sum exdeg indices of unicyclic graphs with given pendant vertices and cycle length were determined.

On the variable sum exdeg index/coindex of graphs

“…Equality in (20) holds if and only if a d1 = • • • = a dn . Therefore, we conclude that equality in (18) holds if and only if G is a regular graph.…”

“…The problem of finding graphs having the extremum values of the variable sum exdeg index of the trees of a fixed order and with the vertices having prescribed degrees was attacked in [11]. Additional recent results about the variable sum exdeg index can be found in the papers [3,7,9,13,18].…”

Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees

“…e extremal result concerning the general sum-connectivity index χ α mentioned in Corollary 1 was proven by using some other way: in [33], for α � 1 and r � 1; in [34,35], for α � 1 and r ≥ 1; in [36], for α > 1 and r ≥ 1. Also, the result concerning SEI a mentioned in Corollary 1 was proven in [37] for r � 1 by other means. Moreover, the result concerning the topological index Pl 2 mentioned in Corollary 2 was proven by using some other way in [38] for r ≥ 1.…”

On the First Three Extremum Values of Variable Sum Exdeg Index of Trees

“…Recently, S. Khalid and A. Ali [12] attempted to find the graphs with the extremal SEI a (a > 1) index value among the trees with prescribed vertex degrees. For more details on the mathematical properties of SEI a index, we refer the readers to [13][14][15][16][17].…”