## About

252

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Introduction

Ismail Naci Cangul currently works at the Faculty of Arts and Science, Bursa Uludag University. Ismail Naci does research in Number Theory, Algebra and Analysis. His current interests are "Studies in Some Algebraic and Topological Properties of Graphs" and "Studies of the topological graph indices, especially for molecular graphs". He recently defined a new graph invariant called Omega in terms of a degree sequence.

Additional affiliations

February 1988 - present

Education

September 1990 - May 1994

September 1989 - June 1990

September 1987 - July 1989

## Publications

Publications (252)

Randić index is one of the most famous topological graph indices. The energy of a graph was defined more than four decades ago for its molecular applications. The classical energy of a graph modeling a molecule is defined as the sum of absolute values of all the eigenvalues of the adjacency matrix corresponding to the modeling graph. There are seve...

We developed a multicriteria decision-making method based on the list of novel single-valued neutrosophic hesitant fuzzy rough (SV-NHFR) weighted averaging and geometric aggregation operators to address the uncertainty and achieve the sustainability of the manufacturing business. In addition, a case study on choosing the optimum elements for a sust...

In graph theory, lattices are used when some structural part of the graph repeats itself finitely or infinitely many times. They have applications in complex analysis and geometry in mathematics, and also natural applications in chemical graph theory. As a lattice can be taken as a graph, it is also possible to use them in the study of large networ...

In this paper, we obtain some upper and lower bounds for the spectral radius of some special matrices such as maximum degree, minimum degree, Randic, sum-connectivity, degree sum, degree square sum, first Zagreb and second Zagreb matrices of a simple connected graph G by the help of matrix theory. We also get some upper bounds for the corresponding...

In this paper, we give some upper and lower bounds for the multiplicative Randic index, reduced reciprocal Randic index, Narumi-Katayama index and symmetric division index a graph using solely the vertex degrees. Then we obtain upper and lower bounds for these indices for the complete graphs, path graphs and Fibonacci-sum graphs. Finally, we compar...

Establishing new relationships between the physical properties and the molecular structure of chemical compounds is very exciting. In this short paper, a QSPR analysis is carried for physical properties of lower alkanes involving Peripheral Wiener index, number of paths of length 3 and the number of vertices in molecular graphs and best multiple li...

A total k-labeling is defined as a function g from the edge set to the first natural number ke and a function f from the vertex set to a non-negative even number up to 2kv, where k = max{ke, 2kv}. A vertex irregular reflexive k-labeling of the graph G is total k-labeling if wt(x) ¹ wt(x¢) for every two different vertices x and x¢ of G, where wt(x)...

e molecular topology of a graph is described by topological indices, which are numerical measures. In theoretical chemistry, topological indices are numerical quantities that are used to represent the molecular topology of networks. ese topological indices can be used to calculate several physical and chemical properties of chemical compounds, such...

Polysaccharides are biomaterial with great biocompatibility, biodegradability, and low toxicity. ere are long chains of monosaccharide units linked together by glycosidic bonds. ey have a wide spectrum of functional properties and are essential to life's survival. ese are a makeup of storage polysaccharides (such as starch and glycogen). Starch is...

Polysaccharides are biomaterial with great biocompatibility, biodegradability, and low toxicity. There are long chains of monosaccharide units linked together by glycosidic bonds. They have a wide spectrum of functional properties and are essential to life’s survival. These are a makeup of storage polysaccharides (such as starch and glycogen). Star...

Recently, some topological graph indices of pentagonal chains and pentagonal double chains are studied and here, we study some topological graph indices of pentagonal triple chains similarly to these previous works. We make use of the vertex and edge partitions of these graphs and calculate their indices by means of these partitions and combinatori...

There are real life situations in our lives where the things are changing continuously or from time to time. It is a very important problem for one whether to continue the existing relationship or to form a new one after some occasions. That is, people, companies, cities, countries, etc. may change their opinion or position rapidly. In this work, w...

In this paper, we present the concept of \(\alpha \)-split domination in graphs. Besides, we calculate the \(\alpha \)-split domination number of path, cycle, complete bipartite graphs and discuss the upper bounds for \(\alpha \)-split domination number in terms of order p, size q and the maximum degree \(\varDelta \). We prove that \(\alpha \)-spl...

In this paper, the determinant of the sum-edge adjacency matrix of any given graph without loops is calculated by means of an algebraic method using spanning elementary subgraphs and also the coefficients of the corresponding sum-edge characteristic polynomial are determined by means of the elementary subgraphs. Also we gave a formula for the numbe...

Energy of a graph, firstly defined by E. Hückel as the sum of absolute values of the eigenvalues of the adjacency matrix, in other words the sum of absolute values of the roots of the characteristic (spectral) polynomials, is an important sub area of graph theory. Symmetry and regularity are two important and desired properties in many areas includ...

In this paper we establish new inequalities involving Randi$\acute{c}$ index, weigthed Randi$\acute{c}$ index and general Randi$\acute{c}$ index in terms of the eigenvalues, the number of edges, the number of vertices, the energy and vertex degrees.

Let vi and vj be two vertices of a graph G. The maximum degree matrix of G is given in [2] by dij = max {di, dj} if vi and vj are adjacent 0 otherwise. Similarly the (i, j)-th entry of the minimum degree matrix is defined by taking the minimum degree instead of the maximum degree above, [1]. In this paper, we have elucidated a relation between maxi...

In this paper, we introduce the Merrifield-Simmons vector defined at a path of corresponding double hexagonal (benzenoid) chain. By utilizing this vector, we present reduction formulae to compute the Merrifield-Simmons index σ(H) of the corresponding double hexagonal (benzenoid) chain H. As the result, we compute σ(H) of H by means of a product of...

In this paper, the concept of accessibility integrity is introduced as a new measure of the stability of a graph $G$ and it is defined as
$$AI(G)=\min\{|S|+m(G-S)\},$$ where $S$ is an accessible set and $m(G-S)$ is the order of a maximum component of $G-S$. First, the accessibility integrity of some graphs is obtained and the relations between acce...

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determine...

Topological graph indices have been of great interest in the research of several properties of chemical substances as it is possible to obtain these properties only by using mathematical calculations. The irregularity indices are the ones to determine the degree of irregularity of a graph. Albertson and Bell indices are two of them. Edge and vertex...

For the vertex set V G of a graph G , the sum of reciprocals of the breadth (distances) between the vertex v ∈ V G and whole other remaining vertices of G is called reciprocal status of v . In this study, first of all, we introduced the V L reciprocal status index and V L reciprocal status co-index of a graph G . Later, we exposed some sharp bounds...

Gutman et al. gave some relations for computing the Hosoya indices of two special benzenoid systems R_n and P_n. In this paper, we compute the Hosoya index and Merrifield-Simmons index of R_n and P_n by means of introducing four vectors for each benzenoid system and index. As a result, we compute the Hosoya index and the Merrifield-Simmons index of...

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its deriv...

An [Formula: see text]-dynamic coloring of a graph [Formula: see text] is a proper coloring [Formula: see text] of the vertices such that [Formula: see text] for each vertex [Formula: see text]. The [Formula: see text]-dynamic chromatic number of a graph [Formula: see text] is the minimum [Formula: see text] such that [Formula: see text] has an [Fo...

Molecules can be modelled by graphs to obtain their required properties by means of only mathematical methods and formulae. In this paper, several degree-based graph indices of one of the important chemical compounds called as polyester are calculated to determine several chemical and physicochemical properties of polyester.

In many areas of science, lattice structures are very useful phenomenons. In network sciences, in chemistry and in social sciences, we face them in the solution of many daily life problems. Several large lattice structures can also be thought as graphs and in that way, are useful in the study of large networks. A very recently defined and studied c...

The stress of a vertex in a graph had been introduced by Shimbel in 1953 as the number of geodesics (shortest paths) passing through it. A topological index of a chemical structure (molecular graph) is a number that correlates given chemical structure with a chemical reactivity or physical property. In this paper, we introduce two new topological i...

An Lh,k-labeling of a graph G=V,E is a function f:V⟶0,∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k. The difference between the highest and lowest assigned values is the index of an Lh,k-labeling. The minimum numb...

A set S ⊆ V of a graph G = V , E is called a co-independent liar’s dominating set of G if (i) for all v ∈ V , N G v ∩ S ≥ 2 , (ii) for every pair u , v ∈ V of distinct vertices, N G u ∪ N G v ∩ S ≥ 3 , and (iii) the induced subgraph of G on V − S has no edge. The minimum cardinality of vertices in such a set is called the co-independent liar’s domi...

In this study, we examine some graph parameters such as the edgenumber, chromatic number, girth, domination number and cliquenumber of power set graphs

Let $G$ be a graph with $n$ vertices and let $d_i$ denote the degree of the vertex $v_i$. For a given graph, there are more than 100 matrices obtained by using some properties of the graph. Most important and used ones are the adjacency, incidency and Laplacian matrices. Recently, several graph topological indices have been used in defining new gra...

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...

Algebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving...

Graph theory is one of the rising areas in mathematics due to its applications in many areas of science. Amongst several study areas in graph theory, spectral graph theory and topological descriptors are in front rows. These descriptors are widely used in QSPR/QSAR studies in mathematical chemistry. Vertex-semitotal graphs are one of the derived gr...

The transmission of a vertex u in a connected graph G is defined as the sum of thedistances between u and all other vertices of a graph G. The reciprocal transmission of a vertex u in a connected graph G is defined as the sum of the reciprocal of distances between u and all other vertices of a graph G. In this paper, we introduce and study new topo...

In this paper, we calculate the Randić type hadi index of some standard graphs, double graphs, subdivision graphs, complements and line graphs. Also we compute the index for the chemical structure graphene.

The eccentric-connectivity index of a graph G is the sum of the products of the eccentricity and the degree of each vertex in G. In this paper, we define four new invariants related to the eccentric-connectivity index and obtain upper bounds for total transformation graphs which are some generalizations of total graph.

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy...

Three main tools to study graphs mathematically are to make use of the vertex degrees, distances and matrices. The classical graph energy was defined by means of the adjacency matrix in 1978 by Gutman and has a large number of applications in chemistry, physics and related areas. As a result of its importance and numerous applications, several modi...

The encryption and decryption is based upon the type of cryptographic scheme being employed and also in some form of key. It is most closely associated with the development and creation of the mathematical algorithms used to encrypt and decrypt messages. The combination of graph labeling techniques together with cryptography to encrypt and decrypt...

Matching number and the spectral properties depending on the characteristic polynomial of a graph obtained by means of the adjacency polynomial has many interesting applications in different areas of science. There are some work giving the relation of these two areas. Here the relations between these two notions are considered and several general r...

Let $G$ be a simple connected graph with vertex set $V(G)$. The status of a vertex $v \in V(G)$ is denoted by $\sigma(v)$ and defined as the sum of distances from $v$ to all other vertices of $G$. The status sum matrix of $G$ is defined by $S_{\sigma}(G) = [s_{ij}]$ where $s_{ij}= \sigma(v_i)+\sigma(v_j)$ if $i \neq j$, and $s_{ij}=0$ otherwise. Th...

Special number sequences play important role in many areas of science. One of them named as Fibonacci sequence dates back to 820 years ago. There is a lot of research on Fibonacci numbers due to their relation with the golden ratio and also due to many applications in Chemistry, Physics, Biology, Anthropology, Social Sciences, Architecture, Anatomy...

Let G be a simple molecular graph without directed and multiple edges and without loops. The vertex and edge-sets of G are denoted by V(G) and E(G), respectively. Suppose G is also a connected molecular graph and let u, v ŒV(G) be two vertices. The harmonic index H(G) of G is defined as the sum of the weights 2(du+dv)⁻¹ of all edges in E(G), where...

The first general Zagreb index \(M^{\alpha }_{1}(G)\) of a graph G is equal to the sum of the \(\alpha \)th powers of the vertex degrees of G. For \(\alpha \ge 0\) and \(k \ge 1\),
we obtain the lower and upper bounds for \(M^{\alpha }_{1}(G)\) and \(M^{\alpha }_{1}(L(G))\) in terms of order, size, minimum/maximum vertex degrees and minimal non-pen...

A recently defined graph invariant denoted by \(\varOmega (G)\) for a graph G is shown to have several applications in graph theory. This number gives direct information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges, etc. In this paper, we use \(\varOmega...

A topological index is a numerical parameter of a graph which characterizes some of the topological properties of the
graph. The concepts of hyper-Zagreb index, �first multiple
Zagreb index, second multiple Zagreb index and relatedly
the Zagreb polynomials were established in chemical graph
theory by means of the vertex degrees. It is reported that...

The algebraic study of graph matrices is an important area of Graph Theory giving information about the chemical and physical properties of the corresponding molecular structure. In this paper, we deal with the edge-Zagreb matrices defined by means of Zagreb indices which are the most frequently used graph indices.

Matching number of a graph is one of the intensively studied areas in graph theory due to numerous applications of the matching and related notions. Recently, Delen and Cangul defined a new graph invariant denoted by Ω which helps to determine several graph theoretical and combinatorial properties of the realizations of a given degree sequence. In...

For a (molecular) graph, the first Zagreb index is equal to the sum of squares of the degrees of vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices. In this paper, we introduce four new tensor products of graphs and study the first and second Zagreb indices and coindices of the res...

Special numbers have very important mathematical properties alongside their numerous applications in many ﬁelds of science. Probably the most important of those is the Fibonacci numbers. In this paper, we use a generalization of Fibonacci numbers called tribonacci numbers having very limited properties and relations compared to Fibonacci numbers. T...

A realizable degree sequence can be realized in many ways as a graph. There are several tests for determining realizability of a degree sequence. Up to now, not much was known about the common properties of these realizations. Euler characteristic is a well-known characteristic of graphs and their underlying surfaces. It is used to determine severa...

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of...

Modelling a chemical compound by a (molecular) graph helps us to obtain some required information about the chemical and physical properties of the corresponding molecular structure. Linear algebraic notions and methods are used to obtain several properties of graphs usually by the help of some graph matrices and these studies form an important sub...

The symmetric figure of a molecule and the fact that the bond polarities are equal imply that the polarities of the bonds cancel each other out and the molecule would be nonpolar. Many molecules are nonpolar, but have polar bonds. A chemical bond is polar if the atoms on either end of its molecular diagram are different. Therefore, the notion of sy...

The ordinary trigonometric functions are defined by means of angles and lengths on the unit circle. There are several derivatives of classical trigonometric functions such as hyperbolic, polar, spherical, Fourier, inverse, log, complex, q-versions etc. In this paper, we add this list a new version of trigonometry which will be consistently called a...

Let G be a simple graph. So called K2 deletion process was recently introduced by Wang. A subgraph G' of G that is obtained as a
result of some K2 deletion process will be called as a crucial subgroup. Let ν(G) and ν(G') be the matching numbers of G and G', respectively. In this study, we study the relation between ν(G), ν(G') and the coefficients...

A graph invariant is a numerical value that depicts the structural properties of an entire graph. The Wiener index is the oldest distance based graph invariant which is defined as the sum of distances between all unordered pair of vertices of the graph G. In this paper we use the method of edge cut to compute the Wiener index and Wiener polarity in...

Recently the second and third authors defined a new graph characteristic similar to the well-known Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined only by means of a given degree sequence, This number gives direct information on the realizability, number of real...

The inverse problem for integer-valued topological indices is about the existence of a graph having its index value equal to a given integer. We solve this problem for the first and second Zagreb indices, and present analogous results also for the forgotten and hyper-Zagreb index. The first Zagreb index of connected graphs can take any even positiv...