Bond incident degree indices of fixed-order unicyclic graphs

<abstract><p>The bond incident degree (BID) index of a graph $ G $ is defined as $ BID_{f}(G) = \sum_{uv\in E(G)}f(d(u), d(v)) $, where $ d(u) $ is the degree of a vertex $ u $ and $ f $ is a non-negative real valued symmetric function of two variables. A graph is stepwise irregular if the degrees of any two of its adjacent vertices differ by exactly one. In this paper, we give a sharp upper bound on the maximum degree of stepwise irregular graphs of order $ n $ when $ n\equiv 2({\rm{mod}}\;4) $, and we give upper bounds on $ BID_{f} $ index in terms of the order $ n $ and the maximum degree $ \Delta $. Moreover, we completely characterize the extremal stepwise irregular graphs of order $ n $ with respect to $ BID_{f} $.</p></abstract>

A connected graph with p vertices and q edges satisfying q=p+1 is referred to as a bicyclic graph. This paper is concerned with an optimal study of the BID (bond incident degree) indices of fixed-size bicyclic graphs with a given matching number. Here, a BID index of a graph G is the number BIDf(G)=∑vw∈E(G)f(dG(v),dG(w)), where E(G) represents G’s edge set, dG(v) denotes vertex v’s degree, and f is a real-valued symmetric function defined on the Cartesian square of the set of all different members of G’s degree sequence. Our results cover several existing indices, including the Sombor index and symmetric division deg index.

We study the general Randić index of a graph [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] is the edge set of [Formula: see text] and [Formula: see text] and [Formula: see text] are the degrees of vertices [Formula: see text] and [Formula: see text], respectively. For [Formula: see text], we present lower bounds on the general Randić index for unicyclic graphs of given diameter and girth, and unicyclic graphs of given diameter. Lower bounds on the classical Randić index and the second modified Zagreb index are corollaries of our results on the general Randić index.

<abstract><p>Recently the exponential Randić index $ {{\rm e}^{\chi}} $ was introduced. The exponential Randić index of a graph $ G $ is defined as the sum of the weights $ {{\rm e}^{{\frac {1}{\sqrt {d \left(u \right) d \left(v \right) }}}}} $ of all edges $ uv $ of $ G $, where $ d(u) $ denotes the degree of a vertex $ u $ in $ G $. In this paper, we give sharp lower and upper bounds on the exponential Randić index of unicyclic graphs.</p></abstract>

Bond incident degree (BID) indices of polyomino chains: A unified approach

The bond incident degree (BID) indices can be written as a linear combination of the number of edges xi,j with end vertices of degree i and j. We introduce two transformations, namely, linearizing and unbranching, on catacondensed pentagonal systems and show that BID indices are monotone with respect to these transformations. We derive a general expression for calculating the BID indices of any catacondensed pentagonal system with a given number of pentagons, angular pentagons, and branched pentagons. Finally, we characterize the CPSs for which BID indices assume extremal values and compute their BID indices.

A tremendous amount of drug experiments revealed that there exists a strong inherent relation between the molecular structures of drugs and their biomedical and pharmacology characteristics. Due to the effectiveness for pharmaceutical and medical scientists of their ability to grasp the biological and chemical characteristics of new drugs, analysis of the bond incident degree (BID) indices is significant of testing the chemical and pharmacological characteristics of drug molecular structures that can make up the defects of chemical and medicine experiments and can provide the theoretical basis for the manufacturing of drugs in pharmaceutical engineering. Such tricks are widely welcomed in developing areas where enough money is lacked to afford sufficient equipment, relevant chemical reagents, and human resources which are required to investigate the performance and the side effects of existing new drugs. This work is devoted to establishing a general expression for calculating the bond incident degree (BID) indices of the line graphs of various well-known chemical structures in drugs, based on the drug molecular structure analysis and edge dividing technique, which is quite common in drug molecular graphs.

On the extremal graphs with respect to bond incident degree indices

Some results on the index of unicyclic graphs

On atom–bond connectivity index of connected graphs

General Randić index of unicyclic graphs with given diameter

Augmented Zagreb index of trees and unicyclic graphs with perfect matchings

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index SEIa and the general zeroth-order Randić index Rα0. The minimum values of SEIa and Rα0 in the class of all trees of fixed order containing no vertex of even degree are obtained for a>1 and α∈[0,1]; also, the maximum value of Rα0 in the mentioned class is determined for 0<α<1. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of SEIa and Rα0 are found for every real number α∉{0,1} and a>1. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of SEIa and Rα0 are determined when a>1 and α does not belong to the closed interval [0,1]; in the same class, the maximum values of Rα0 are also found for 0<α<1. The graphs that achieve the obtained extremal values are also determined.

Numerous molecular structure descriptors, which may be used in theoretical chemistry, are the bond incident degree (BID) indices. This study is devoted to establish a general expression for calculating the BID indices of pentagonal chains and to find the extremal (maximal and minimal) values for a variety of BID indices over the certain collection of pentagonal chains with a fixed number of pentagons.

This article gives bounds on a substantial number of BID (bond incident degree) indices for connected graphs in terms of their order, size, and maximum degree. The considered BID indices include, among others, the Sombor index (together with its reduced version), atom-bond sum-connectivity index, symmetric division deg index, sum-connectivity index, harmonic index, and Randi´c index. All the graphs that attain the obtained bounds are also characterized. All the established bounds are valid also for molecular graphs. A graph of order n and size m is called an (n,m)-graph. The obtained bounds provide a partial solution to the problem of finding graphs with extremum (considered) BID indices over the class of all connected (n,m)-graphs with a fixed maximum degree under certain constraints.