Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

A bond incident degree (BID) index of a graph G is defined as ∑ u v ∈ E G f d G u , d G v , where d G w denotes the degree of a vertex w of G , E G is the edge set of G , and f is a real-valued symmetric function. The choice f d G u , d G v = a d G u + a d G v in the aforementioned formula gives the variable sum exdeg index SEI a , where a ≠ 1 is any positive real number. A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by V n , k the class of all n -vertex graphs with k ≥ 1 cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of SEI a from the set V n k for a > 1 . In the present paper, we not only characterize the graphs with the minimum value of SEI a from the set V n k for a > 1 , but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n -vertex molecular graphs with k ≥ 1 cut vertices and containing at least one cycle.

Five results on maximizing topological indices in graphs

In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.

Investigation of General Power Sum-Connectivity Index for Some Classes of Extremal Graphs

In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

Extremal Graphs on the Rupture Degree

For a given graph [Formula: see text], by [Formula: see text] and [Formula: see text] denote the order of the largest component and the number of connected components of [Formula: see text], respectively. The rupture degree of [Formula: see text] is an important combinatorial parameters, which defined as [Formula: see text]. Clearly, the smaller the value [Formula: see text] is, the better the connectivity of graph [Formula: see text] is. In this paper, the tree structures with minimum rupture degree be discussed, and the extremal graphs on the rupture degree in terms of graph girth are also characterized.