Roslan Hasni

• Professor at University of Malaysia, Terengganu

Let be a simple connected graph with vertex set and edge set . The Randić index of graph is the value , where and refer to the degree of the vertices and . We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.

Let G be a simple graph. A function ϕ:V(G)→{1,2,…,k} a vertex k-labeling which assigns labels to the vertices of G. For any edge xy in G, we define the weight of this edge as wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is termed as an edge irregular k -labeling of G. The smallest possible value of k for which the graph G possesse...

After the Chartrand definition of graph labeling, since 1988 lot of graph families have been labeled through mathematical techniques. A basic approach in those labeling was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018 Asim applied computer-based algorithms to overcome this limitation and label...

The first Zagreb index of graphs is defined to be the sum of squares of degrees of all the vertices of graphs. It drew a great deal of attention in the past half-century. In this paper, we study the relationship between the first Zagreb index and Roman domination number of graphs. More precisely, we characterize the trees with maximum the first Zag...

The harmonic index of graph [Formula: see text] is the value [Formula: see text], where [Formula: see text] refers to the degree of [Formula: see text]. Zhong [The harmonic index for graphs, Appl. Math. Lett. 25 (2012) 561–566] proved that [Formula: see text] for any tree [Formula: see text] of order [Formula: see text]. As a results of Ali, Raza a...

In graph theory, the concept of domination is essential in a variety of domains. It has broad applications in diverse fields such as coding theory, computer network models, and school bus routing and facility location problems. If a fuzzy graph fails to obtain acceptable results, neutrosophic sets and neutrosophic graphs can be used to model uncert...

Circular intuitionistic fuzzy sets (CIFS) are a recent extension of intuitionistic fuzzy sets (IFS) that can handle imprecise membership values effectively. However, its representation is limited to the space under the intuitionistic fuzzy interpretation triangle (IFIT). To address this, a new generalization of CIFS called circular q-rung orthopair...

Let G be a molecular graph. The atom-bond connectivity (ABC) and geometric-arithmetic (GA) indices of G are defined as and , where (or ) denoted the degree of the vertex u (or v), respectively. A dendrimer is a hyperbranched molecule built up from branched units called monomers. In this paper, the ABC and GA indices for two families of dendrimer na...

There is a strong correlation between the properties of their molecular structure based on medical experiments. The molecular structure of drugs contains all of the information for investigating their chemical, physical and biological properties. These properties can be determined using a theoretical descriptor tool known as topological indices. In...

The Randić index is among the most famous degree-based topological indices in chemical graph theory. It was introduced due to its application in modeling the properties of certain molecular structures and has been extensively studied. In this paper, we study the lower bound of the Randić index of trees in terms of the order and the total domination...

p>Let G = (V (G), E(G)) be a graph, define an edge labeling function ψ from E(G) to {0, 1, . . . , k − 1} where k is an integer, 2 ≤ k ≤ |E(G)|, induces a vertex labeling function ψ∗ from V (G) to {0, 1, . . . , k − 1} such that ψ∗(v) = ψ(e1) × ψ(e2) × . . . × ψ(en) mod k where e1, e2, . . . , en are all edge incident to v. This function ψ is calle...

With respect to a simple graph G, a vertex labeling ϕ: V(G) > {1,2,...,k) is known as k-labeling. The weight corresponding to an edge xy in G, expressed as wϕ (xy), represents the labels sum of end vertices x and y, given by wϕ (xy) = ϕ(x) + ϕ(y) A vertex k-labeling is expressed as an edge irregular k-labeling with respect to graph G provided that...

For a graph $ G $, we define a total $ k $-labeling $ \varphi $ is a combination of an edge labeling $ \varphi_e(x)\to\{1, 2, \ldots, k_e\} $ and a vertex labeling $ \varphi_v(x) \to \{0, 2, \ldots, 2k_v\} $, such that $ \varphi(x) = \varphi_v(x) $ if $ x\in V(G) $ and $ \varphi(x) = \varphi_e(x) $ if $ x\in E(G) $, then $ k = \, \mbox{max}\, \{k_e...

The first Zagreb index of graphs is defined to be the sum of squares of degrees of all the vertices of graphs. It drew a great deal of attention in the past half-century. In this paper, we study the relationship between the first Zagreb index and Roman domination number of graphs. More precisely, we characterize the trees with the maximum first Zag...

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe(x) → {1, 2,. .. , ke} and a vertex labeling ϕv(x) → {0, 2,. .. , 2kv}, such that ϕ(x) = ϕv(x) if x ∈ V (G) and ϕ(x) = ϕe(x) if x ∈ E(G), where k = max {ke, 2kv}. The total k-labeling ϕ is called an edge irregular reflexive k-labeling of G, if for every two edges x...

The topological index is a molecular predictor that is commonly supported in the research of QSAR of pharmaceuticals to numerically quantify their molecular features. Entropic network measures are a class of topological descriptors with applications that range from quantitative characterization of a molecular structure to the study of certain chemi...

One of the vertex-degree based topological indices is Sombor index which is denoted by SO(G), and defined by SO(G)=∑_(uv∈E(G)) √(d_u^2+d_v^2 ) where d_u,d_v are the degree of vertices u and v in the graph G respectively. In this paper, we are focusing on computing the Sombor index of some graph operations, more precisely join and corona product of...

In this paper, we discuss a case-study of personal sharing of information among students of two undergraduate programme, i.e., Computational Mathematics and Software Engineering (in short, CM and SE respectively) at UniversitiMalaysia Terengganu, Malaysia. The data collected is represented as a directed graph with edges between vertices representin...

In a graph G, the distance between two vertices is the length of the shortest path between them. The maximum distance between a vertex to any other vertex is considered as the eccentricity of the vertex. In this paper, we introduce the first general Zagreb eccentricity index and found upper and lower bounds on this index in terms of order, size and...

A bijective map ρ from V(Ω) →{1,2, … |V(Ω)|} is called sum divisor cordial labeling for graph Ω so that for every uυ ∈ E(Ω) edge is fixed the label 1 if 2 divides ρ (u) + ρ(υ) and 0 otherwise, then the difference between number of edges labeled with 1 and the number of edges labeled with 0 by at most 1. A graph is called sum divisor cordial graph i...

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. Recently, there has been a lot of interest in graph energy research. Previous literature has suggested integrating energy, Laplacian energy, and signless Laplacian energy with single-valued neutrosophic graphs (SVNGs). This integration is used to solve problems t...

A convex polytope is the convex hull of a finite set of points in the Euclidean space R n. By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. e detour index is defined t...

A convex polytope is the convex hull of a finite set of points in the Euclidean space R n. By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. e detour index is defined t...

Neutrosophic set (NS) is a framework used when the imprecision and uncertainty of an event are described based on three possible aspects, ie, the membership degree, neutral membership degree and non-membership degree. On the other hand, neutrosophic graphs (NG) are applicable to deal with bulk information events. Furthermore, the incidence graph co...

We define a total k-labeling φ of a graph G as a combination of an edge labeling φe:E(G)→{1,2,…,ke} and a vertex labeling φv:V(G)→{0,2,…,2kv}, such that φ(x)=φv(x) if x∈V(G) and φ(x)=φe(x) if x∈E(G), where k= max {ke,2kv}. The total k-labeling φ is called an edge irregular reflexive k-labeling of G if every two different edges has distinct edge wei...

Let R be a commutative ring. The Von.-Neuman regular (Shortly Vn. Neum. reg.) graph of R _ is a graph which its vertices are all items of R s. t. thither is an edge between vertices a, b if a+b is a Vn.-Neum. reg. item of R. Here a new definition of the Vn. Neum. reg. graph of R called pseudo–Vn.–Neum. reg. graph of R denoted by P-VG(R) is a graph...

Let G be a simple graph with vertex set V(G) and edge set E(G). The atom-bond connectivity and Randić index of graph G are defined as and , respectively, where du denotes the degree of vertex u in G. Let ABC – R denotes the difference between ABC index and R index. In this paper, we present a further ordering for the ABC – R indices, and determine...

A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to find the boiling point of paraffin and total π-electron ene...

Embeddings are often viewed as a high-level representation of systematic methods to simulate an algorithm designed for one kind of parallel machine on a different network structure and/or techniques to distribute data/program variables to achieve optimum use of all available processors. A topological index is a numeric quantity of a molecule that i...

This paper deals with decomposition of complete graphs on n vertices into circulant graphs with reduced degree r < n-1. They are denoted as Cn(a1, a2,..., am), where a1 to am are generators. Mathematical labeling for such bigger (higher order and huge size) and complex (strictly regular with so many triangles) graphs is very difficult. That is why...

An edge-magic total (EMT) labeling for graph Γ is one-one map from π : V(Γ) ∪ E(Γ) → {1, 2, … ,|V(Γ)|+|E(Γ)|}, so that there manage a number c along a rule that for every edge, uv ∈ E(Γ), π (u) + π (uv) + π (v) = c. And if all vertices are assigned with positive integral numbers {1, 2, … ,|V(Γ)|} then this type of labeling is called a super EMT lab...

Graph labeling is an assignment of (usually) positive integers to elements of a graph (vertices and/or edges) satisfying certain condition(s). In the last two decades, graph labeling research received much attention from researchers. This articles is about edge irregularity strength for some classes of plane graphs. Edge irregularity strength denot...

For a simple graph G, a vertex labeling : V (G) ! f1; 2; : : : ; kg is called k-labeling. The weight of an edge xy in G, denoted by w�(xy), is the sum of the labels of end vertices x and y, i.e. w (xy) = (x)+ (y). A vertex k-labeling is de ned to be an edge irregular k-labeling of the graph G if for every two di�erent edges e and f, there is w (e)...

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and t...

Let G be a graph of order n with vertices labeled as v1, v2, . . . , vn. Let di be the degree of the vertex vi , for i � 1, 2, . . . , n. 'e difference adjacency matrix of G is the square matrix of order n whose (i, j) entry is equal to ( ��������� di + dj − 2 􏽱 − 1)/( ���� didj 􏽱 ) if the vertices vi and vj of G are adjacent or (vivj ∈ E(G)) and z...

A reverse edge magic total (REMT) labeling for the graph Γ = (V(Γ), E(Γ)) along with cardinality of p = |V(G)| and q = |E(G)| respectively. A one-one map π: V(Γ) ∪ E(Γ) → {1, 2, … ,|V(Γ)|+|E(Γ)|} having a rule that for every edge, uv ϵ E(Γ), then π (uv) − {π (u) + π (v)} = c, where c is a magic number. A reverse edge magic total (REMT) labeling is...

Let us consider a mapping [Formula: see text] of a graph [Formula: see text], where [Formula: see text] is an integer, [Formula: see text]. The mapping [Formula: see text] induces for every vertex [Formula: see text] of [Formula: see text] the label [Formula: see text]. Let [Formula: see text] ([Formula: see text]) denote the number of edges (verti...

Degree based topological indices of chemical structures of the conductive 2D MOFs Cu 3(HITP)2[m, n] are studied. Expressions for multiple Zagreb indices and Zagreb polynomials for these consequential classes of networks are obtained.

In this paper, we investigate the cordiality of the isomorphic copies of paths mPn . We also give sufficient condition for mPn to admit the prime cordial labeling, product cordial labeling and total product cordial labeling. Furthermore, we also determine the prime cordial labeling, product cordial labeling and total product cordial labeling for su...

Let be a graph with vertices and be the degree of its -th vertex ( is the degree of ). In this article, we compute the generalization of Zagreb index, the generalized Zagreb index, the first and second hyper -indices, the sum connectivity F-index, and the product connectivity F-index graphs of , , and . 1. Introduction Mathematical chemistry is a...

Let G be a simple graph with vertex set V (G) and edge set E(G). The geometric-arithmetic index (GA index for short) of graph G is defined as GA(G) = P uv∈E(G) 2 √ dudv du+dv , where the summation extends over all edges uv of G, and du denotes the degree of vertex u in G. Recently, Du et al. [On geometric arithmetic indices of (molecular) trees, un...

The Optical Transpose Interconnection System (OTIS) has applications in parallel processing, distributed processing, routing, and networks. It is used for efficient usage of multiple parallel algorithms or parallel systems, with different global interconnections in a network as it is an optoelectronic (combination of light signals and electronics)....

A graph G with n vertices is called a k–step Hamiltonian graph if the n vertices can be labeled as the sequence υ1, υ2,…,υn such that d(υi, υi+1) = d(υ1, υn) = k for each i = 1,2,…, n − 1. In this paper, we continue the study of k–step Hamiltonian graphs and obtain several properties.

Graph ϒ = (V(ϒ), E(ϒ)) contain finite nodes V(ϒ) and finite edges E(ϒ). We also represent the order of the graph and size as μ = |V(ϒ)| and v = |V(ϒ)|. A graph is called (a, d)-edge magic total (EAT) labeling if there exists a bijective map φ from V(ϒ) ∪ E(ϒ) to the elements if the weight-set X = {ω(qr)|qr ∈ E(ϒ)} is arithmetic progression (A. P.)...

A solitary number that can be utilized to describe some property of the graph of a particle is known as a topological index for that graph. There are various topological indices that have discovered a few applications in hypothetical science. In QSAR/QSPR think about, physico-concoction properties and topological indices, for example, Randic¢, atom...

There are various teaching methods developed in order to attain successful delivery of a subject without prior knowledge of the interaction among the students in a class. Social network analysis (SNA) can be used to identify individual, intermediate and group measures of interaction in a classroom. The idea is on identifying ways to boost the stude...

Motivated by the definitions of the irregular labeling of a graph defined by Chartrand et al. [10] in 1988 and Baˇca et al. [8] in 2007, Ahmad et al. [5] in 2014 define the edge irregular k-labeling of a graph. In this paper, we show that several types of trees, namely non-homogeneous caterpillar, homogeneous lobster and homogenous amalgamation sta...

There are various teaching methods developed in order to attain successful delivery of a subject without prior knowledge of the interaction among the students in a class. Social network analysis can be used to identify individual, intermediate and group measures of interaction in a classroom. The idea is on identifying ways to boost the students pe...

Let G be a graph with vertex set V (G) and edge set E(G). A (p, q)-graph G = (V, E) is said to be AL(k)-traversal if there exists a sequence of vertices (v1, v2,. .. , vp) such that for each i = 1, 2,. .. , p − 1, the distance between vi and vi+1 is k. We call a graph G a k-step Hamiltonian graph (or say it admits a k-step Hamiltonian cycle) if it...

For a given integer k, a given graph G on n vertices is called k-step Hamil-tonian (or just k-SH) if the vertices of G can be labeled as v1, v2, ..., vn such that d(v1, vn) = k and d(vi, vi+1) = k for each i = 1, 2, ..., n − 1. In this paper, we present a construction namely B-construction that produces a (k+i)-SH graph from any k-SH graph G for ev...

For a graph G = (V, E) we define a labeling : V (G) ! {1, 2, · · · , k} to be a vertex irregular k-labeling. A vertex irregular k-labeling is defined to be an edge irregular k-labeling of the graph G = (V, E) if every two different edges vu and v0u0 satisfy their weight w (vu) = w (v) + w (u) 6= w (v0) + w (v0) = w (v0u0). The minimum k for which t...

Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this paper we discus...

For a graph G = (V, E) we define a labeling : V (G) ! {1, 2, · · · , k} to be a vertex irregular k-labeling. A vertex irregular k-labeling is defined to be an edge irregular k-labeling of the graph G = (V, E) if every two different edges vu and v0u0 satisfy their weight w (vu) = w (v) + w (u) 6= w (v0) + w (v0) = w (v0u0). The minimum k for which t...

The Randić index R(G) of a graph G is the sum of the weights (d u d v ) − 1/2 of all edges uv in G, where d u denotes the degree of vertex u. Du and Zhou [On Randić indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760–2770] determined the n-vertex trees with the third for n ≥ 7, the fourth for n ≥ 10, the...

Topological descriptors are numerical parameters of a molecular graph which characterize its molecular topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical comp...

For a simple graph G, a vertex labeling ϕ:V(G)→{1,2,…,k} is called a vertex k-labeling. For any edge xy in G, its weight wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is called an edge irregular k-labeling of G. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denot...

A (p, q) −graph G with vertex set V(G) and edge set E(G) is said to be AL(k) −traversable for k ≥ 1 if we can arrange its vertex set as the sequence of vertices {v1,v2, …, vp} such that the distance between vi and vi+1 for each i = 1,2, …, p −1 is k. A graph G is called k −step Hamiltonian if it is AL(k) −traversable and d(v1,vp) = k. Then, the seq...

For a graph G, let P(G, λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent (or simply χ-equivalent), denoted by G ∼ H, if P(G, λ) = P(H, λ). A graph G is chromatically unique (or simply χ-unique) if for any graph H such as H ∼ G, we have H ≅ = G, i.e, H is isomorphic to G. In this paper, the chromatic uniquenes...

For a graph G, let P(G,λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent (or simply χ-equivalent), denoted by G∼ H, if P(G,λ)=P(H,λ). A graph G is chromatically unique (or simply χ-unique) if for any graph H such as H∼ G, we have H≅ G, i.e, H is isomorphic to G. In this paper, the chromatic uniqueness of a new...

For a given integer k, a graph G of order n is called k-step Hamiltonian if there is a labeling v 1 , v 2 , ..., v n of vertices of G such that d(v 1 , v n) = d(v i , v i+1) = k for i = 1, 2, ..., n − 1. The independence number of a graph is the maximum cardinality of a subset of pair-wise non-adjacent vertices. In this paper we study the independe...

Algorithms help in solving many problems, where other mathematical solutions are very complex or impossible. In this paper edge irregularity strength of a complete graph es(Kn) is computed using the algorithm that is impossible to compute manually on higher order graphs. Using the values of es(Kn) an upper-bound is suggested that is far better than...

The neighbourhood polynomial N(G,x) is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph and it is defined as , where N(G) is neighbourhood complex of a graph, whose vertices of the graph and faces are subsets of vertices that have a common neighbour. A dendrimers is an artificially manufactured...

Let P(G, λ) be the chromatic polynomial of a graph G. Two graphs G and H are said to be chromatically equivalent, denoted G ∼ H, if P(G, λ) = P(H, λ). We write [G] = {H|H ∼ G}. If [G] = {G}, then G is said to be chromatically unique. In this paper, we first characterize certain complete 6-partite graphs with 6n+1 vertices according to the number of...

Algorithms help in solving many problems, where other mathematical solutions are very complex or impossible. Computations help in tackling numerous issues, where other numerical arrangements are extremely perplexing or incomprehensible. In this paper, the edge irregularity strength of a complete binary tree (T2,h), complete ternary tree (T3,h) and...

The atom bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as \( ABC(G) = \sum\nolimits_{uv \in E(G)} {\sqrt {d_{v} + d_{u} - 2/d_{v} \cdot d_{u} } } \), where d u denotes the degree of a vertex u in G. In t...

The atom-bond connectivity (ABC) index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as ABC(G) =∑ uv∈E(G) √[dv +du -2]/[dvdu], where du denotes the degree of a vertex u in G. In this paper, we establish the general formulas for the a...

For such a graph G, suppose P(G, λ) denote the chromatic polynomial of graph G. Let G and H are two graphs, then G and H are said to be chromatically equivalent (or simply χ - equivalent) denoted by G ~ H, if P(G, λ) = P(H, λ). A graph G is said to be chromatically unique (or simply χ -unique) if for any graph H such that G ~ H, we have G H, that i...

In this paper we consider the cordiality of a generalized Jahangir graph $J_{n,m}$. We give sufficient condition for $J_{n,m}$ to admit (or not admit) the prime cordial labeling, product cordial labeling and total product cordial labeling.

For a graph G, suppose P(G,l) be the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent (or simply χ-equivalent), denoted by G ~ H, if P(G, λ)= P(H, λ). A graph G is said to be chromatically unique (or simply χ-unique) if for any graph H such that H ~ G, we have H ≅ G, i.e. H is isomorphic to G. In this paper, the...

Pentacyclic graphs with maximum Estrada index

For a graph G = (V (G), E(G)), an edge labeling function f : E(G) → {0, 1, · · · , k − 1} where k is an integer, 2 ≤ k ≤ |E(G)|, induces a vertex labeling function f*: V (G) → {0, 1, · · · , k − 1} such that f*(v) is the product of the labels of the edges incident to v (mod k). This function f is called k-total edge product cordial (or simply k-TEP...

The atom-bond connectivity index of a graph G, denoted as ABC(G), is defined as the sum of the weight sqrt((du+dv-2)/dudv)of all edges uv of G, where du (or dv) denotes the degree of vertex u (or v) in G. The ABC index provides a good model for the stability of linear and branches alkanes as well as the strain energy of cycloalkanes. In this note,...

For a graph G, let P(G; λ) denote the chromatic polynomial of G. Two graphs G and H are chromatically equivalent if they share the same chromatic polynomial. A graph G is chromatically unique if any graph chromatically equivalent to G is isomorphic to G. A K4- homeomorph is a subdivision of the complete graph K4. In this paper, we determine a famil...

Let G be a simple connected molecular graph. The eccentric connectivity index \( \upxi(G) \) is defined as \( \xi (G) = \mathop \sum \nolimits_{v\epsilon V(G)} { \text{deg} } (v)\,\text{ec}(v) \), where \( { \text{deg} }(v) \) denotes the degree of vertex v and \( {\text{ec}}(v) \) is the largest distance between v and any other vertex \( u\,\epsil...

There are certain types of topological indices such as degree-based topological indices, distance-based topological indices and counting-related topological indices. Among degree-based topological indices, the so-called atom-bond connectivity (ABC), geometric-arithmetic (GA) are of vital importance. These topological indices correlate certain physi...

The geometric-arithmetic index (GA index for short) is a newly proposed graph invariant, based on the end-vertex degrees of all edges of a graph, in mathematical chemistry. Du et al. [On geometric arithmetic indices of (molecular) trees, unicyclic graphs and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011), 681-697] determined the first...

Let G be a molecular graph. The distance between two vertices of G is the length of a shortest path connecting these two vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex in G. In this paper, we consider some infinite families of molecular graphs with application to cycloalkanes and compute their s...

The index of atom-bond connectivity (ABC) models the stability of linear and branched alka- nes,uv as well as the strain energy of cycloalkanes. This index is defined as ABC(G) = ∑ uv∈E(G) ( d v + d u −2)/( d v × d u ) The degree of the vertex u is the number of edges with u as an end vertex, denotes by du in G. In this work, we compute the ABC ind...

Let G = (V,E) be a simple connected molecular graph. The eccentric connectivity index ξ (G) is a distance-based molecular structure descriptor that was recently used for mathematical modelling of biological activities of diverse nature. In such a simple molecular graph, vertices represent atoms and edges represent chemical bonds, we denoted the set...

The atom-bond connectivity (ABC) index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as ABC(G) =∑ uv∈E(G) √ [dv +du -2]/[dv.du], where du denotes the degree of a vertex u in G. In this paper, we establish the general formulas for the...

In this paper, we investigate the new graph characteristic, the edge irregularity strength, denoted as es, as a modification of the well known irregularity strength, total edge irregularity strength and total vertex irregularity strength. As a result, we obtain the exact value of an edge irregularity strength of corona product of cycle with isolate...

A topological index of graph G is a numerical parameter related to G which characterizes its molecular topology and is usually graph invariant. In the field of quantitative structure-activity (QSAR)/quantitative structure-activity structure-property (QSPR) research, theoretical properties of the chemical compounds and their molecular topological in...

The atom-bond connectivity (ABC) index is one of the newly most studied degree based molecular structure descriptors, which have chemical applications. For a graph G, the ABC index can be defined as A B C ( G ) = Σ u v ∈ E ( G ) d v + d u − 2 / d v . d u , where d u , the degree of the vertex u is the number of edges with u as an end v...

Let G = (V, E) be a simple connected molecular graph. In such a simple molecular graph, vertices and edges are depicted atoms and chemical bonds respectively, we refer to the sets of vertices by V (G) and edges by E (G). If d(u, v) be distance between two vertices u, v ∈ V(G) and can be defined as the length of a shortest path joining them. Then, t...

For a graph $G$, let $P(G,\lambda)$ denote the chromatic polynomial of $G$. Two graphs $G$ and $H$ are chromatically equivalent if they share the same chromatic polynomial. A graph $G$ is chromatically unique if for any graph chromatically equivalent to $G$ is isomorphic to $G$. In this paper, the chromatically unique of a new family of 6-bridge gr...