Extremal graphs with respect to variable sum exdeg index via majorization

Discrete Applied Mathematics

Dimitrov

2018

Many existing degree based topological indices can be clasified as bond incident degree (BID) indices, whose general form iswhere uv is the edge connecting the vertices u, v of the graph G, E(G) is the edge set of G, d u is the degree of the vertex u and Ψ is a non-negative real valued (symmetric) function of d u and d v . Here, it has been proven that if the extension of Ψ to the interval [0, ∞) satisfies certain conditions then the extremal (n, m)-graph with respect to the BID index (corresponding to Ψ) must contain at least one vertex of degree n − 1. It has been shown that these conditions are satisfied for the general sum-connectivity index (whose special cases are: the first Zagreb index and the Hyper Zagreb index), for the general Platt index (whose special cases are: the first reformulated Zagreb index and the Platt index) and for the variable sum exdeg index. Applying aforementioned result, graphs with maximum values of the aforementioned BID indices among tree, unicyclic, bicyclic, tricyclic and tetracyclic graphs were characterized. Some of these results are new and the already existing results are proven in a shorter and more unified way. arXiv:1707.00733v1 [math.CO]

Section: 5mentioning

confidence: 68%

“…Remark 2.13. Recently, Ghalavand and Ashrafi [12] determined the graphs with maximum SEI a value for a > 1 among all the n-vertex trees, unicyclic, bicyclic and tricyclic graphs. They claimed that the graph G 3 has the maximum SEI a value for a > 1 among tricyclic graphs.…”

Section: 5mentioning

confidence: 99%

On the extremal graphs with respect to bond incident degree indices

Discrete Applied Mathematics

Dimitrov

2018

“…Using the definition of SEI a , Yarahmadi and Ashrafi [15] introduced the variable sum exdeg polynomial and explored some mathematical aspects of this polynomial. For a > 1, Ghalavand and Ashrafi [7] solved some extremal problems for SEI a using the concept of majorization. Ali and Dimitrov [2] gave alternative proofs of four main results proved in [7] and extended one of the theorems proved in [7].…”

Section: Introductionmentioning

confidence: 99%

“…For a > 1, Ghalavand and Ashrafi [7] solved some extremal problems for SEI a using the concept of majorization. Ali and Dimitrov [2] gave alternative proofs of four main results proved in [7] and extended one of the theorems proved in [7]. Khalid and Ali [10] investigated the properties of the variable sum exdeg index SEI a of trees having a fixed number of vertices and vertices with a prescribed degree.…”

Section: Introductionmentioning

confidence: 99%

On the extremal cactus graphs for variable sum exdeg index with a fixed number of cycles

Javaid

AKCE International Journal of Graphs and Combinatorics

Milovanovic

et al. 2020

The variable sum exdeg index, introduced by Vuki cevi c [Croat. Chem. Acta 84 (2011) 87-91] for predicting the octanol-water partition coefficient of certain chemical compounds, of a graph G is defined as SEI a ðGÞ ¼ P v2VðGÞ d v a dv , where a is any positive real number different from 1, V(G) is the vertex set of G and d v denotes the degree of a vertex v. A connected graph G is a cactus if and only if every edge of G lies on at most one cycle. For n > 3 and k ! 0, let C n, k be the class of all n-vertex cacti with k cycles. The present paper is devoted to find the graphs with minimal and maximal SEI a values among all the members of the graph class C n, k for a > 1.

“…The topological index SEI a is very well correlated with octanol-water partition coefficient of octane isomers [27]. Detail about the chemical applicability and mathematical properties of this index can be found in the references [2,9,27,28,30]. A vertex having degree 1 is called pendent vertex and a vertex which have degree greater than 2 is named as branching vertex.…”

Section: Introductionmentioning

confidence: 99%

On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

Khalid

Discrete Math. Algorithm. Appl.

2018

The zeroth-order general Randić index (usually denoted by R 0 α ) and variable sum exdeg index (denoted by SEI a ) of a graph G are defined as R 0, a is a positive real number different from 1 and α is a real number other than 0 and 1. A segment of a tree is a path P , whose terminal vertices are branching or pendent, and all non-terminal vertices (if exist) of P have degree 2. For n ≥ 6, let PT n,n1 , ST n,k , BT n,b be the collections of all n-vertex trees having n 1 pendent vertices, k segments, b branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections PT n,n1 , ST n,k , BT n,b . The obtained extremal trees for the collection ST n,k are also extremal trees for the collection of all n-vertex trees having fixed number of vertices with degree 2 (because it is already known that the number of segments of a tree T can be determined from the number of vertices of T with degree 2 and vise versa).