Szeged Index Research Articles

Trees with minimum weighted Szeged index

Trees with minimum weighted Szeged index

Szeged Index and Padmakar-Ivan Index of Nilpotent Graph of Integer Modulo Ring with Prime Power Order

Recently, graphs have started to be used to represent a finite ring. Nikmehr and Khojasteh in the article defined the nilpotent graph of a ring . Denoted , is a graph with the set of vertices being all the elements in the ring Two vertices and are adjacent if and only if is nilpotent elements in the ring . Topological index is a field that discusses graph structure based on the degree of each vertex of a graph and the distance between vertices. In this study, the author will gives the general formula of the Szeged index and Padmakar-Ivan index of the nilpotent graph graph of the modulo ring with prime power order. The result of this research is a general formula for the topological indices of nilpotent graphs of the integer modulo ring, called the Szeged index and the Padmakar-Ivan index.

Szeged index of hollow hexagons

AbstractThe Szeged index of a connected graph () is a well known distance based topological index. A primitive coronoid system is a coronoid system formed by a single chain in a macro‐cyclic arrangement consisting of linearly and angularly annelated hexagons. The angular hexagons are called corners. A hollow hexagon is a primitive coronoid system with exactly six corners. In this paper we calculate the values of Szeged index of hollow hexagons using the cut method.

ON THE SZEGED INDEX AND ITS NON-COMMUTING GRAPH

In chemistry, the molecular structure can be represented as a graph. Based on the information from the graph, its characterization can be determined by computing the topological index. Topological index is a numerical value that can be computed by using some algorithms and properties of the graph. Meanwhile, the non-commuting graph is a graph, in which two distinct vertices are adjacent if and only if they do not commute, where it is made up of the non-central elements in a group as a vertex set. In this paper, the Szeged index of the non-commuting graph of some finite groups are computed. This paper focuses on three finite groups which are the quasidihedral groups, the dihedral groups, and the generalized quaternion groups. The construction of the graph is done by using Maple software. In finding the Szeged index, some of the previous results and properties of the graph for the quasidihedral groups, the dihedral groups, and the generalized quaternion groups are used. The generalisation of the Szeged index of the non-commuting graph is then established for the aforementioned groups. The results are then applied to find the Szeged index of the non-commuting graph of ammonia molecule.

On Molecular Structural Characterization of Cyclen Cored Dendrimers

Macromolecules are gaining much attention in various fields today. Dendrimers are artificially synthesized macromolecules by convergent or divergent approach. They are compact regular structures with spherical dimension and has a vast number of application in disparate fields such as drug delivery, material science, and biology, magnetic resonance imaging, an organic light-emitting device, etc. Determining the pharmacological, chemical, and biological characteristics of a substance necessitates a significant amount of effort. From the chemical graph of the dendrimer structure, we can infer those characteristics with the help of numerical descriptors known as the topological index. The Wiener and Szeged indices are two important distance-based topological indices applicable in nanoscience. The degree-based topological indices also have great importance and huge applications in structural chemistry. These indices together with graph entropy are found to be more effective and have found application in different sciences. In this work, the Wiener index, Szeged indices, Mostar indices, and Padmakar Ivan index for cyclen cored dendrimers are evaluated by converting the original graph into quotient graphs using Θ * - classes. This technique is applied since the regular cut method is a lengthy process while moving on to higher generations and also due to the presence of odd cycles in the structure. The degree-based indices and the degree-based graph entropies for the cyclen cored dendrimers are further studied. The comparison graphs with respect to the topological indices as well as graph entropies are also presented.

Comparing Wiener, Szeged and revised Szeged index on cactus graphs

We show that on cactus graphs the Szeged index is bounded above by twice the Wiener index. For the revised Szeged index the situation is reversed if the graph class is further restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index is bounded below by twice its Wiener index. Additionally, we show that these bounds are sharp and examine the cases of equality. Along the way, we provide a formulation of the revised Szeged index as a sum over vertices, which proves very helpful, and may be interesting in other contexts.

The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

<abstract><p>Let $ H $ be a connected graph. The edge revised Szeged index of $ H $ is defined as $ Sz^{\ast}_{e}(H) = \sum\limits_{e = uv\in E_H}(m_{u}(e|H)+\frac{m_{0}(e|H)}{2})(m_{v}(e|H)+\frac{m_{0}(e|H)}{2}) $, where $ m_{u}(e|H) $ (resp., $ m_{v}(e|H) $) is the number of edges whose distance to vertex $ u $ (resp., $ v $) is smaller than to vertex $ v $ (resp., $ u $), and $ m_{0}(e|H) $ is the number of edges equidistant from $ u $ and $ v $. In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized.</p></abstract>

On the relationship between variable Wiener index and variable Szeged index

We resolve two conjectures of Hriňáková et al. (2019)[10] concerning the relationship between the variable Wiener index and variable Szeged index for a connected, non-complete graph, one of which would imply the other. The strong conjecture is that for any such graph there is a critical exponent in (0,1], below which the variable Wiener index is larger and above which the variable Szeged index is larger. The weak conjecture is that the variable Szeged index is always larger for any exponent exceeding 1. They proved the weak conjecture for bipartite graphs, and the strong conjecture for trees.In this note we disprove the strong conjecture, although we show that it is true for almost all graphs, and for bipartite and block graphs. We also show that the weak conjecture holds for all graphs by proving a majorization relationship.

Topological indices of the wreath product of graphs

Topological indices, i.e., numerical invariants suitably associated to graphs and only depending upon their isomorphism type, have important applications in Chemistry. Their computation constitutes an important branch of Chemical Graph Theory. In this paper, we focus on some degree and distance-based invariants related to the Zagreb indices, the Szeged index and the Wiener index, namely, the F-index, the vertex PI index and the hyper-Wiener index. In particular, we find the formulæ to compute these topological invariants for wreath product of graphs.

Computation of edge Pi index, vertex Pi index and Szeged index of some cactus chains

A cactus chain is a connected graph in which all blocks are cycles, each cycle has at most two cut-vertices and each cut-vertex is shared by exactly two cycles. In this paper we give exact values of edge PI index and vertex PI index of an arbitrary cactus chain and vertex Szeged index of some special types of cactus chains.

On Szeged‐type indices of titanium oxideTiO2nanotubes

AbstractThe relations and correlation between the physicochemical characteristics of a chemical compounds and the molecular structure connectivity of the same compound are used in quantitative structure activity and property relationship studies (QSAR/QSPR) by using the appropriate graph‐theoretical techniques. Szeged index has been extensively studied for modeling physicochemical properties of organic compounds acting as drugs or possess pharmacological activity. In this paper, we consider the two‐dimensional lattices of titanium oxide nanotubes TiO2. We compute several Szeged‐type indices such as vertex Szeged, edge Szeged, edge‐vertex Szeged, total Szeged, Padmaker–Ivan, revised Szeged and revised‐edge Szeged indices of these graphs. We define new types of cuts that are different from orthogonal cuts and can be used to obtain Szeged‐type results of titania nanotubes.

On revised Szeged index of a class of unicyclic graphs

Computing topological indices of graphs is a fundamental and classical topic. Let [Formula: see text] be a connected graph. The revised Szeged index [Formula: see text] is defined as [Formula: see text], where [Formula: see text] (respectively, [Formula: see text]) is the number of vertices whose distance to vertex [Formula: see text] (respectively, [Formula: see text]) is smaller than the distance to vertex [Formula: see text] (respectively, [Formula: see text]), and [Formula: see text] is the number of vertices equidistant from both ends of [Formula: see text]. In this paper, we determine the smallest revised Szeged index among all conjugated unicyclic graphs (i.e., unicyclic graphs with perfect matchings), and the corresponding extremal graphs are characterized.

On vertex PI index of certain triangular tessellation networks

Abstract The Wiener index, due to its many applications is considered to be one of very important distance-based index. But the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges. The PI index like other distance related indices has great disseminating power. The index was firstly investigated by Khadikar et al. (2001), they have probed the chemical applications of the PI index. They proved that the proposed PI index correlates highly with the physicochemical properties and biological activities of a large number of diverse and complex chemical compounds and the Wiener and Szeged indices. Recently, the vertex Padmarkar-Ivan (PIv) index of a chemical graph G was introduced as the sum over all edges uv of a molecular graph G of the vertices of the graph that are not equidistant to the vertices u and v. In this paper, the vertex PIv index of certain triangular tessellation are computed by using graph-theoretic analysis, combinatorial computing, and edge-dividing technology.

Vertex Szeged indices of P 2n + F P n +1

Let G=(V,E) be a simple connected graph, where V(G) and E(G) represent the vertex set and edge set of G respectively. For a graph the vertex Szeged index is equal to the product over all edges uv of G of the number of vertices which are not equidistant to vertices u and v. The vertex Padmarker-Ivan (PIv ) index of a graph is the sum over all edges uv of G of the number of vertices which are not equidistant to vertices u and v. The aim of this paper is to compute and compare the vertex Szeged index and vertex Padmarker-Ivan (PIv ) index of P 2n+F Pn +1, where P 2n+F Pn +1 represents four operation on P 2n ×Pn +1.

三圈图的极大边修正Szeged指标

三圈图的极大边修正Szeged指标

An inequality between variable wiener index and variable szeged index

A well-known inequality between the Szeged and Wiener indices says that Sz(G)=∑e=ij∈E(G)ne(i)ne(j)≥∑{u,v}d(u,v)=W(G) for every graph G. In the past, variable variations of the standard topological indices were defined. Following this line, we study a natural generalisation of the above inequality, namely ∑e=ij∈E(G)(ne(i)ne(j))α≥∑{u,v}d(u,v)α. We show that for all trees the inequality is true if α > 1, and the opposite inequality holds if 0 ≤ α < 1. In fact, the first result also holds for bipartite graphs and for graphs on n vertices with at most n+3 edges, but the opposite one does not. For general graphs we solve also the case α < 0 and we present interesting conjectures. Observe, that both the sums are interesting on their own, and in accordance with the usual terminology they can be called the variable Szeged and variable Wiener indices.

Comparison between Szeged indices of graphs

The Szeged index Sz(G) of a simple connected graph G is the sum of the terms nu (e)nv (e) over all edges e = uv of G, where nu (e) is the number of vertices of G lying closer to u than v, and nv (e) is defined analogously. The aim of this paper is to present some relationship between Szeged index and some of its variants such as the edge-vertex Szeged index, the vertex-edge Szeged index and revised Szeged index. Moreover, we obtain lower and upper bounds on the difference between vertex-edge Szeged index and edge-vertex Szeged index of unicyclic graphs.

Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes

Let I be a summation-type topological index. The I-complexity CI (G) of a graph G is the number of different contributions to I(G) in its summation formula. In this paper the complexity CSz(G) is investigated, where Sz is the well-studied Szeged index. Let Oe(G) (resp. Ov(G)) be the number of edge (resp. vertex) orbits of G. While CSz(G)  Oe(G) holds for any graph G, it is shown that for any m  there exists a vertex-transitive graph Gm with CSz(Gm) = Oe(Gm) = m. Also, for any \le k \le m + 1 there exists a graph Gm,k with CSz(Gm,k) = Oe(Gm,k) = m and CW(Gm,k) = Ov(Gm,k) = k. The Sz-complexity is determined for a family of (5,0)-nanotubical fullerenes and the Szeged index is compared with the total eccentricity.

Further Remarks on n-Distance-Balanced Graphs

Throughout this paper, we present a new strong property of graph so-called nicely n-distance-balanced which is notably stronger than the concept of n-distance-balanced recently given by the authors. We also initially introduce a newgraph invariant which modies Szeged index and is suitable to study n-distance-balanced graphs. Looking for the graphs extremal with respect to the modiedSzeged index it turns out the n-distance-balanced graphs with odd integer n arethe only bipartite graphs which can maximize the modied Szeged index and thisalso disproves a conjecture proposed by Khalifeh et al. [Khalifeh M.H.,Youse-Azari H., Ashra A.R., Wagner S.G.: Some new results on distance-based graphinvariants, European J. Combin. 30 (2009) 1149-1163]. Furthermore, we gathersome facts concerning with the nicely n-distance-balanced graphs generated by somewell-known graph products. To enlighten the reader some examples are provided.Moreover, a conjecture and a problem are presented within the results of this article.

Minimum Szeged index among unicyclic graphs with perfect matchings

Let G be a connected graph. The Szeged index of G is defined as $$Sz(G)=\sum \nolimits _{e=uv\in E(G)}n_{u}(e|G)n_{v}(e|G)$$ , where $$n_{u}(e|G)$$ (resp., $$n_{v}(e|G)$$ ) is the number of vertices whose distance to vertex u (resp., v) is smaller than the distance to vertex v (resp., u), and $$n_{0}(e|G)$$ is the number of vertices equidistant from both ends of e. Let $$\mathcal {M}(2\beta )$$ be the set of unicyclic graphs with order $$2\beta $$ and a perfect matching. In this paper, we determine the minimum value of Szeged index and characterize the extremal graph with the minimum Szeged index among all unicyclic graphs with perfect matchings.