Computation of Polynomial Degree-Based Topological Descriptors of Indu-Bala Product of Two Paths

Cheminformatics is entirely a newly coined term that encompasses a field that includes engineering computer sciences along with basic sciences. As we all know, vertices and edges form a network whereas vertex and its degrees contribute to joining edges. The degree of vertex is very much dependent on a reasonable proportion of network properties. There is no doubt that a network has to have a reliance of different kinds of hub buses, serials, and other connecting points to constitute a system that is the backbone of cheminformatics. The Indu-Bala product of two graphs G 1 and G 2 has a special notation as described in Section 2. The attainment of this product is very much due to related vertices at to different places of G 1 ∨ G 2 . This study states we have found M-polynomial and degree-based topological indices for Indu-Bala product of two paths P k and P j for j , k ≥ 2 . We also give some graphical representation of these indices and analyzed them graphically.

New Results on the Geometric-Arithmetic Index

Let G be a graph with vertex set V G and edge set E G . Let d u denote the degree of vertex u ∈ V G . The geometric-arithmetic index of G is defined as GA G = ∑ u v ∈ E G 2 d u d v / d u + d v . In this paper, we obtain some new lower and upper bounds for the geometric-arithmetic index and improve some known bounds. Moreover, we investigate the relationships between geometric-arithmetic index and several other topological indices.

Minimum Size Highly Redundantly Rigid Graphs in the Plane

A graph G is said to be k-vertex rigid in $$\mathbb {R}^d$$ R d if $$G-X$$ G - X is rigid in $$\mathbb {R}^d$$ R d for all subsets X of the vertex set of G with cardinality less than k. We determine the smallest number of edges in a k-vertex rigid graph on n vertices in $$\mathbb {R}^2$$ R 2 , for all $$k\ge 4$$ k ≥ 4 . We also consider k-edge-rigid graphs, defined by removing edges, as well as k-vertex globally rigid and k-edge globally rigid graphs in $$\mathbb {R}^d$$ R d . For $$d=2$$ d = 2 we determine the corresponding tight bounds for each of these versions, for all $$k\ge 3$$ k ≥ 3 . Our results complete the solutions of these extremal problems in the plane. The result on k-vertex rigidity verifies a conjecture of Kaszanitzky and Király (Graphs Combin, 32:225–240, 2016). We also determine the degree of vertex redundancy of powers of cycles, with respect to rigidity in the plane, answering a question of Yu and Anderson (Int J Robust Nonlinear Control, 19(13):1427–1446, 2009).

On Sombor Index

The concept of Sombor index (SO) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index SO: SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of vertex vi in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work.

Bounds of modified Sombor index, spectral radius and energy

<abstract><p>Let $ G $ be a simple graph with edge set $ E(G) $. The modified Sombor index is defined as $ ^{m}SO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_{u}^{2}~~+~~d_{v}^{2}}} $, where $ d_{u} $ (resp. $ d_{v} $) denotes the degree of vertex $ u $ (resp. $ v $). In this paper, we determine some bounds for the modified Sombor indices of graphs with given some parameters (e.g., maximum degree $ \Delta $, minimum degree $ \delta $, diameter $ d $, girth $ g $) and the Nordhaus-Gaddum-type results. We also obtain the relationship between modified Sombor index and some other indices. At last, we obtain some bounds for the modified spectral radius and energy.</p></abstract>

Ordering chemical graphs by Sombor indices and its applications

Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as where denotes the degree of vertex in . In this paper, our aim is to order the chemical trees, chemical unicyclic graphs, chemical bicyclic graphs and chemical tricyclic graphs with respect to Sombor index and reduced Sombor index. We determine the first fourteen minimum chemical trees, the first four minimum chemical unicyclic graphs, the first three minimum chemical bicyclic graphs, the first seven minimum chemical tricyclic graphs. At last, we consider the applications of reduced Sombor index to octane isomers.

The general Albertson irregularity index of graphs

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G $ and define it as $ A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where $ p $ is a positive real number and $ d(v) $ is the degree of the vertex $ v $ in $ G $. The new index is not only generalization of the well-known Albertson irregularity index and $ \sigma $-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.</p></abstract>

On Extremal Graphs of Degree Distance Index by Using Edge-Grafting Transformations Method

Background: Topological indices have numerous implementations in chemistry, biology and in lot of other areas. It is a real number associated to a graph, which provides information about its physical and chemical properties and their correlations. For a connected graph H, the degree distance defined as DD(H)=∑_(\h_1,h_2⊆V(H))〖(〖deg〗_H (h_1 )+〖deg〗_H (h_2 )) d_H (h_1,h_2 ) 〗, where 〖deg〗_H (h_1 ) is the degree of vertex h_1and d_H (h_1,h_2 ) is the distance between h_1and h_2in the graph H. Aim and Objective: In this article, we characterize some extremal trees with respect to degree distance index which has a lot of applications in theoretical and computational chemistry. Materials and Methods: A novel method of edge-grafting transformations is used. We discuss the behavior of DD index under four edge-grafting transformations. Results: By the help of those transformations, we derive some extremal trees under certain parameters including pendant vertices, diameter, matching and domination numbers. Some extremal trees for this graph invariant are also characterized. Conclusion: It is shown that balanced spider approaches to the smallest DD index among trees having given fixed leaves. The tree Cn,d has the smallest DD index, among the all trees of diameter d. It is also proved that the matching number and domination numbers are equal for trees having minimum DD index.

On the maximum ABC index of bipartite graphs without pendent vertices

For a simple graph G, the atom–bond connectivity index (ABC) of G is defined as ABC(G) = $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$where d(v) denotes the degree of vertex v of G. In this paper, we prove that for any bipartite graph G of order n ≥ 6, size 2n − 3 with δ(G) ≥ 2, $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$and we characterize all extreme bipartite graphs.