# Akbar AliUniversity of Hail · Department of Mathematics

Akbar Ali

PhD

## About

127

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Introduction

Currently, I'm an associate professor with the Department of Mathematics, University of Hail, Hail, Saudi Arabia. My research interests include Discrete Mathematics, Combinatorics, Mathematical Chemistry, especially (Chemical) Graph Theory. I'm serving as a member of the editorial boards of the journals "MATCH Communications in Mathematical and in Computer Chemistry", "Communications in Combinatorics and Optimization", and some others.

Additional affiliations

October 2021 - present

September 2019 - September 2021

September 2016 - September 2019

**University of Management and Technology**

Position

- Professor (Assistant)

Education

January 2012 - February 2016

September 2009 - June 2011

## Publications

Publications (127)

Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is $BID(G)=\sum_{uv\in E(G)}$ $\Psi(d_{u},d_{v})$, where $uv$ 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 $\Psi$ is a non-negative real...

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The Mostar index is a recently introduced bond-additive distance-based graph invariant that measures the degree of peripherality of particular edges and of the graph as a whole. It attracted considerable attention, both in the context of complex networks and in more classical applications of chemical graph theory, where it turned out to be useful a...

Die Theorie der Regularen Graphen (The Theory of Regular Graphs), written by the Danish Mathematician Julius Petersen in 1891, is often considered the first strictly theoretical paper dealing with graphs. In the 130 years since then, regular graphs have been a common and popular area of study. While regular graphs are typically considered to be gra...

Let G be a graph containing no component isomorphic to the path graph of order 2. Denote by du the degree of an arbitrary vertex u of G. The augmented Zagreb index (AZI) of G is the sum of the weights (du.dv /(du + dv − 2))^3 over all edges uv of G. In this note, the unique graph with minimal AZI is characterized from the class of all connected tri...

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected $k$-cyclic graph. Abdo, Dimitrov, and Gutman [Discrete Appl. Math. 250 (2018) 57-64] characterized the graphs having...

For a connected graph G on at least three vertices, the augmented Zagreb index (AZI) of $G$ is defined as $$AZI(G)=\sum_{uv\in E(G)}\left(\frac{d(u)d(v)}{d(u)+d(v)-2}\right)^{3},$$ being a topological index well-correlated with the formation heat of heptanes and octanes. A k-apex tree G is a connected graph admitting a k-subset $X\subset V(G)$ such...

For a graph G, its bond incident degree (BID) index is defined as the sum of the contributions f(du , dv) over all edges uv of G, where dw denotes the degree of a vertex w of G and f is a real-valued symmetric function. If f (du , dv) = du + dv or du*dv then the corresponding BID index is known as the first Zagreb index M1 or the second Zagreb inde...

The cyclomatic number of a graph $G$, denoted by $\nu(G)$ or simply by $\nu$, is the minimum number of those edges of $G$ whose removal make $G$ as acyclic. Denote by $\mathbb{G}_{n,\nu}$ the class of all $n$-vertex connected graphs with cyclomatic number $\nu$. The problems of finding graph(s) possessing the maximum second Zagreb index $M_2$ for $...

The general sum-connectivity index is a molecular descriptor introduced within the field of mathematical chemistry about a decade ago. For an arbitrary real number α, the general sum-connectivity index of a graph G is denoted Xα(G) and is defined as the sum of the numbers (d(u) + d(v))^α over all edges uv of G, where d(u) and d(v) denote the degree...

The graphs having the maximum value of certain bond incident degree indices (including the second Zagreb index, general sum-connectivity index, and general zeroth-order Randić index) in the class of all connected graphs with fixed order and number of pendent vertices are characterized in this paper. The problem of finding graphs having the minimum...

A bond incident degree (BID) index of a graph G is defined as the sum of the quantities f(d_G(u), d_G(v)) over all pairs of adjacent vertices u, v of G, where d_G(w) denotes the degree of the vertex w of G, and f is a real-valued symmetric function. This paper reports extremal results for a special type of BID indices. The obtained results extend a...

The molecular descriptors that are defined in terms of connection numbers of adjacent vertices of the chemical graph of the considered chemical structure are known as bond incident connection (BIC) indices. The primary aim of this article is to study BIC indices of polyomino and benzenoid chains.

A connected graph in which no edge lies on more than one cycle is called a cactus graph (also known as Husimi tree). A bond incident degree (BID) index of a graph G is defined as the sum of the outputs f(du, dv) over all edges of G, where dw denotes the degree of a vertex w of G and f is a real-valued symmetric function. This study involves extrema...

Background:
A topological index of a molecular graph is the numeric quantity that can be used to predict certain physical or/and chemical properties of the corresponding molecule. The first and second multiplicative Zagreb indices are the topological indices introduced by Gutman about a decade ago. Xu et al. used some graph transformations which in...

Let G be a connected graph with the vertex set V = {v1, v2, . . ., vn}, where n ≥ 2. Denote by di the degree of the vertex vi for i = 1, 2, . . . , n. If vi and vj are adjacent in G, we write i = j, otherwise we write i sim; j. The variable sum exdeg index and coindex of G are defined as SEIa(G) = P ij(adi + adj ) = Pn i=1 diadi and SEIa(G) = Σ i∼j...

For a graph G, its bond-additive indices are defined as \sum_{uv∈E(G)} β(u, v), where E(G) is the edge set of G and β is a real-valued function satisfying the property β(u, v) = β(v, u). By atoms-pair-additive indices of a graph G, we mean graph invariants of the form \sum_{u,v∈V(G)} α(u, v)/2, where V(G) is the vertex set of G and α is a real-valu...

A set S of vertices in a connected graph G of diameter d is an irregular dominating set if it is possible to assign distinct labels from the set {1, 2, . . . , d} to the vertices of S in such a way that for every vertex v of G, there exists a vertex u of S such that the distance from v to u is the label of u. If exactly two vertices of S are permit...

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] cha...

The multiplicative first Zagreb index of a graph H is defined as the product of the squares of the degrees of vertices of H . The line graph of a graph H is denoted by L H and is defined as the graph whose vertex set is the edge set of H where two vertices of L H are adjacent if and only if they are adjacent in H . The multiplicative first Zagreb i...

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G, its vertex-degree-based topological indices of the form BIDG=∑uv∈EGβdu,dv are known as bond incident degree indices, where EG is the edge set of G, dw denotes degree of an arbitrary vertex w of G, and β is a real-valued-symmetric function. Thos...

Let G be a graph containing no isolated vertices. For the graph G, its modified first Zagreb index is defined as the sum of reciprocals of squares of vertex degrees of G. This article provides some new bounds on the modified first Zagreb index of G in terms of some other well-known graph invariants of G. From the obtained bounds, several known resu...

The connective eccentricity index (CEI) of a connected graph G is defined as \(\xi ^{ee}(G)=\sum _{u\in V_G}[d_G(u)/\varepsilon _G(u)]\), where \(d_G(u)\) and \(\varepsilon _G(u)\) are the degree and eccentricity, respectively, of the vertex \(u\in V_G\) of G. In this paper, graphs with the maximum CEI are characterized from the class of all connec...

An edge coloring c of a graph G is a royal k-edge coloring of G if the edges of G are assigned nonempty subsets of the set {1, 2, ... , k} in such a way that the vertex coloring obtained by assigning the union of the colors of the incident edges of each vertex is a proper vertex coloring. If the vertex coloring is vertex-distinguishing, then c is a...

This paper is concerned with three recently introduced degree-based graph invariants; namely, the Sombor index, the reduced Sombor index, and the average Sombor index. The first aim of the present paper is to give some results that may be helpful in proving a recently proposed conjecture concerning the Sombor index. Establishing inequalities relate...

The Sombor index is a recently introduced graph-theoretical invariant of the bond-additive type. It is known that it takes integer values for bipartite semi-regular graphs whose degrees appear as two smaller elements in a Pythagorean triple. In this note we show that it can have integer values also for graphs with more complicated structure and con...

The atom-bond connectivity (ABC) index was introduced in the last quarter of the 1990s to improve the prediction power of the Randić index. Later on, in 2008, the factor √ 2 was dropped from the original definition of the ABC index, and some additional chemical applications of this index were reported, which resulted in considerable interest in stu...

Let G be a graph containing no component isomorphic to the path graph of order 2. Denote by dw the degree of a vertex w in G. The augmented Zagreb index (AZI) of G is the sum of the quantities (du*dv /(d u + d v − 2))^3 over all edges uv of G. Denote by G(n, χ) the class of all connected graphs of a fixed order n and with a fixed chromatic number χ...

The first and second Zagreb indices are the molecular descriptors that were appeared within a study of molecular modeling more than four decades ago. The coindex versions of these Zagreb indices were introduced around a decade before. Finding trees with the first two extremum Zagreb indices/coindices from the class of all chemical trees of a fixed...

The Wiener polarity index Wp is a topological index that was devised by the chemist Harold Wiener for predicting the boiling points of alkanes. The index Wp for chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices at distance 3. A vertex of a chemical tree with degree at least 3 is called a b...

The Wiener polarity index Wp of a graph is defined as the number of unordered pairs of its vertices at distance 3. The problem of finding trees attaining the maximum Wp value, among all chemical trees of a fixed order n, was solved in the paper [Mol. Inf. 38 (2019) Art# 1800076] for n ≥ 8. Motivated by the usage of Wp in a recent publication [J. Ch...

In this chapter the concept of irregular graphs is looked at in another way, namely by re-interpreting what is meant by the degree of a vertex.

It was seen in the preceding chapter that every connected graph of order 3 or more has an irregular weighting. Therefore, if G is a connected graph of order 3 or more, there exists a weighting w : E(G) → [k] for some integer k ≥ 2 such that the vertices in the resulting weighted graph H of G have distinct degrees. To obtain an irregular weighted gr...

We saw in Chap. 1 that it is impossible for the degrees of every two vertices of a nontrivial graph G to be different. However, rather than considering all vertices of G, if one were to consider the vertices individually and investigate the degrees of the neighbors or the structure of the subgraph induced by the neighbors of a vertex, an entirely d...

Among the many results in graph theory dealing with graph decompositions are those involving complete graphs. While the primary emphasis has been graph decompositions in which every two subgraphs are the same (isomorphic), here we consider graph decompositions in which not only are every two subgraphs non-isomorphic, but they possess some prescribe...

An Eulerian circuit in a connected graph G is a circuit that contains every edge of G exactly once while an Eulerian walk in G is a closed walk that contains every edge of G at least once. While only Eulerian graphs contain an Eulerian circuit, every nontrivial connected graph contains an Eulerian walk. The irregularity concept here is an irregular...

In 1985, Frank Harary and Michael Plantholt introduced another way to obtain an irregular labeling (or weighting or coloring) of the vertices of a graph by assigning integers of the set [k] to the edges of the graph. For each vertex v, rather than adding or averaging the colors of the edges incident with v, they assigned the set of colors of the ed...

In this chapter, the concept of irregular graphs is looked at in another way, by considering multigraphs rather than graphs or, equivalently, by considering weighted graphs.

This paper is concerned with a recently introduced graph invariant, namely the Sombor index. Some bounds on the Sombor index are derived and then utilized to establish additional bounds by making use of the existing results. One of the direct consequences of one of the obtained bounds is that the cycle graph Cn attains the minimum Sombor index amon...

The Wiener polarity index of a graph G, usually denoted by Wp(G), is defined as the number of unordered pairs of those vertices of G that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree T is a non-trivial path S whose end-vertices have degrees different from 2 in T and every other verte...

The augmented Zagreb index (AZI) is a molecular structure descriptor intro- duced about a decade ago. It is defined as AZI = Σ uv ( du dv du + dv - 2 )3 ; where du is the degree of the vertex u, and the summation goes over all pairs of adjacent vertices of the underlying molecular graph. Chemical applicability of AZI was tested in several studies,...

The general sum-connectivity index of a graph $G$ is defined as $\chi_{\alpha}(G)= \sum_{uv\in E(G)} (d_u + d_{v})^{\alpha}$ where $d_{u}$ is degree of the vertex $u\in V(G)$, $\alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $\chi_...

The modified Albertson index, denoted by A* of a graph G is defined as the sum of the quantities |du^2- dv^2|, where du, dv denote the degrees of the vertices u, v, respectively, of G and E(G) is the edge set of G. In this note, a sharp lower bound of A* in terms of the maximum degree for the case of trees is derived. The n-vertex trees having maxi...

The modified first Zagreb connection index ZC1* for a graph G is defined as ZC1*(G) = [Formula]. A branching vertex of a graph is a vertex with degree greater than 2. In this paper, graphs with the maximum and minimum ZC1* values are characterized from the class of all trees of a fixed order and having a fixed number of branching vertices.

The variable sum exdeg index is a graph invariant introduced in 2011 for the purpose of predicting a particular physicochemical property, namely the octanol-water partition coefficient, of certain molecules. The variable sum exdeg coindex is devised by making some modifications in the definition of the variable sum exdeg index. The primary aim of t...

The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph G can be defined as MγG=∑uv∈EGdGγ−1u+dGγ−1v, where γ is a real number. As MγG is equal to the order and size of G when γ=0 and γ=1, respectively, γ is usually assumed to be different from 0 to 1. In this paper, for every integer γ≥2, the FGZ index...

Let $G=(V,E)$ be a simple connected
graph with $n$ vertices, $m$ edges and sequence of vertex degrees
$d_1 \ge d_2 \ge \cdots \ge d_n>0$, $d_i=d(v_i)$, where $v_i\in V$. With $i\sim j$ we denote adjacency of
vertices $v_i$ and $v_j$. The general
sum--connectivity index of graph is defined as $\chi_{\alpha}(G)=\sum_{i\sim j}(d_i+d_j)^{\alpha}$, wher...

The variable sum exdeg index SEI_a was introduced by Vukičević [Croat. Chem. Acta 84 (2011) 87-91] for predicting the octanol-water partition coefficient of certain chemical compounds, where is "a" is a positive real number different from 1. A connected graph G is a cactus if and only if every edge of G lies on at most one cycle. For n ≥ 4 and k ≥...

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n, size m with vertex degree sequence ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(vi). Denote by G a complement of G. If vertices vi and v j are adjacent in G, we write i ~ j, otherwise we write i j. The general zeroth-order Randic coindex of ' G is defined as 0Ra(G) = ∑i j (d a-1...

The set of all different degrees of the vertices of a graph G is known as the degree set of G. A nontrivial graph of order n whose degree set consists of n−1 elements is called an antiregular graph. Antiregular graphs have been studied in literature also under other names, including "quasi-perfect graphs", "maximally nonregular graphs" and "degree...

An $n$-vertex graph whose degree set consists of exactly $n-1$ elements is called antiregular graph. Such type of graphs are usually considered opposite to the regular graphs. An irregularity measure ($IM$) of a connected graph $G$ is a non-negative graph invariant satisfying the property: $IM(G) = 0$ if and only if $G$ is regular. The total irregu...

In this paper, the graphs having sixth to fifteenth maximum harmonic indices are characterized from the class of all n-vertex trees for sufficiently large n.
Here is the link for the full text of this paper: https://rdcu.be/b5Rxp

Let CT_{n,k} and CT*_{n,b} be the classes of all n-vertex chemical trees with k segments and b branching vertices, respectively, where 3 ≤ k ≤ n − 1 and 1 ≤ b < n/2 − 1. The solution of the problem of finding trees from the class CT_{n,k} or CT*_{n,b}, with the minimum first Zagreb index or minimum second Zagreb index follows directly from the main...

The connectivity index, introduced by the chemist Milan Randic in 1975, is one of the topological indices with many applications.
In the first quarter of 1990s, Randic proposed the variable connectivity index by extending the definition of the connectivity index.
The variable connectivity index for graph G is defined as \sum_{vw∈E(G)}((d(v) + c)(d(...

The variable connectivity index, introduced by the chemist Milan Randic in the first quarter of the 1990s, for a graph G is defined as \sum_{vw∈E(G)}((dv + c)(dw + c))^{-1/2}, where c is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randic index. This paper extends the recent study [S. Yousaf...

Zagreb indices are among the most studied molecular structure descriptors. In this paper, we present upper bounds on the first Zagreb index, second Zagreb index, second multiplicative Zagreb index, multiplicative sum Zagreb index and a lower bound on the first multiplicative Zagreb index for simple connected molecular (n, m)-graphs, which are the s...

The modified first Zagreb connection index ZC1* is a graph invariant that was appeared
about fifty years ago within a study of molecular modeling, and after a long time it has
been revisited in the two papers [A. Ali, N. Trinajstic, Mol. Inform. 37(6-7), (2018) Art#
1800008; arXiv:1705.10430 [math.CO] (2017)] and [A. M. Naji, N. D. Soner, I. Gutman...

Energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the absolute values of the eigenvalues of $G$. Two graphs with the same order and energy are called equienergetic graphs. A graph $G$ with the property $G\cong \overline{G}$ is called self-complementary graph, where $\overline{G}$ denotes the complement of $G$. Two non-self-co...

Bell's degree-variance Var$\!{}_{B}$ for a graph $G$, with the degree sequence ($d_1,d_2,\ldots,d_n$) and size $m$, is defined as
$Var\!_{B} (G)=\frac{1}{n} \sum _{i=1}^{n}\left[d_{i} -\frac{2m}{n}\right]^{2}$.
In this paper, a new version of the irregularity measures of variance-type, denoted by $Var_q$, is introduced and discussed. Based on a com...

Let $G=(V,E)$ be a simple connected graph with the vertex set $V=\{1,2,\ldots,n\}$ and sequence of vertex degrees ($d_1,d_2,\cdots,d_n$) where $d_i$ denotes the degree of a vertex $i\in V$. With $i\sim j$, we denote the adjacency of the vertices $i$ and $j$ in the graph $G$. The inverse sum indeg ($ISI$) index of the graph $G$ is defined as $ISI(G)...

The symmetric division deg (SDD) index is one of the 148 discrete Adriatic indices, introduced several years ago. The SDD index has already been proved a valuable index in the QSPR/QSAR (quantitative structure-property/activity relationships) studies. In the present paper, we firstly correct an upper bound on the SDD index of molecular trees, repor...

Let $G$ be a simple connected non-trivial graph of order $n$, size $m$, and vertex degree sequence ($d_1, d_2,\cdots,d_n$). The first Zagreb index $M_1$, forgotten index $F$ and inverse degree $ID$ are the graph invariants defined as $M_1(G)=\sum_{i=1}^n d_i^2$, $ F(G) = \sum_{i=1}^n d_i^3$ and $ID(G)=\sum_{i=1}^n \frac{1}{d_i}$, respectively. A gr...

The modified first Zagreb connection index ZC*1 is a graph invariant, initially appeared within a formula of the total electron energy of alternant hydrocarbons in 1972, and revisted in a recent paper [A. Ali, N. Trinajstić. A novel/old modification of the first Zagreb index. Mol Inform. 2018;37(6). Art# 1800008; arXiv:1705.10430 [math.CO]]. In thi...

Three general molecular descriptors, namely the general sum‐connectivity index, general Platt index and ordinary generalized geometric‐arithmetic index, are studied here. Best possible bounds for the aforementioned descriptors of arbitrary saturated hydrocarbons are derived under certain constraints. These bounds are expressed in terms of number of...

Let CT_{n,k} and CT_{n,b} be the classes of all n-vertex chemical trees with k segments and b branching vertices, respectively. The solution of the problem of finding trees from the class CT_{n,k} or CT_{n,b}, with the minimum first Zagreb index or minimum second Zagreb index follows directly from the main results of [MATCH Commun. Math. Comput. Ch...

The cyclomatic number $\nu$ of a graph G is the minimum number of those edges of G whose removal makes G as acyclic.
The second Zagreb index $M_2$ for a graph G is the sum of the products of degrees of adjacent vertices of G.
For $\nu = \binom{k-1}{2}+t$ with $1 \le t \le k-1$ and $4 \le k \le n-2$, let $G^*$ be the graph having maximum $M_2$ value...

The general sum-connectivity index of a graph $G$, denoted by $\chi_{_\alpha}(G)$, is defined as $\sum_{uv\in E(G)}(d(u)+d(v))^{\alpha}$, where $uv$ is the edge connecting the vertices $u,v\in V(G)$, $d(w)$ denotes the degree of a vertex $w\in V(G)$, and $\alpha$ is a non-zero real number. For $\alpha=-1/2$ and $n\geq 11$, Wang \textit{et al.} [On...

An edge coloring c of a graph G is a royal k-edge coloring of G if the edges of G are assigned nonempty subsets of the set {1, 2,. .. , k} in such a way that the vertex coloring obtained by assigning the union of the colors of the incident edges of each vertex is a proper vertex coloring. If the vertex coloring is vertex-distinguishing, then c is a...

Energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the absolute values of the eigenvalues of $G$. Two graphs with the same order and energy are called equienergetic graphs. A graph $G$ with the property $G\cong \overline{G}$ is called self-complementary graph, where $\overline{G}$ denotes the complement of $G$. Two non-self-co...

The neighborhood first Zagreb index (NM1) has been recently introduced for characterizing the topological structure of molecular graphs. In this study, we present some sharp bounds on the index NM1 and establish its relations with the first and second Zagreb indices in case of some special graphs. It is verified and demonstrated on examples that in...

A novel concept is outlined by which the total irregularity irr t (G), introduced recently by Abdo and Dimitrov, can be extended. It is demonstrated on examples that starting with this concept several generalized versions of the total irregularity can be established.

It is well known fact that several physicochemical properties of chemical compounds are closely related to their molecular structure. Mathematical chemistry provides a method to predict the aforementioned properties of compounds using topological indices. The Zagreb indices are among the most studied topological indices. Recently, three modified ve...

Following Estrada [The Structure of Complex Networks , Oxford University Press, Oxford, UK, 2011], we use the word "network" interchangeably with "graph" in this paper. Several classes of particular nonregular graphs have been introduced in literature; for example, the classes of n-vertex connected antiregular graphs, highly irregular graphs, total...

The modified first Zagreb connection index (ZC_1^*) is a molecular descriptor, which was initially appeared within a formula of the total electron energy of alternant hydrocarbons in 1972. In a recent paper [A. Ali, N. Trinajstic, A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO] (2017); Mol. Inform. 37 (2018) 1800008],...

The present study is devoted to characterize the cactus with minimum T1 index and maximum T2 index over the class of all cacti having fixed number of vertices and cycles, where T1 index is the first multiplicative Zagreb index, Narumi-Katayama index, modified second Zagreb index, or harmonic index, and T2 index is the second multiplicative Zagreb i...