Szeged Index Research Articles

On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and η(G)=Sz(G)−W(G), where W(G) and Sz(G) are the Wiener and Szeged indices of G, respectively. A well-known result of Klavžar, Rajapakse, and Gutman states that η(G)≥0, and by a result of Dobrynin and Gutman η(G)=0 if and only if each block of G is complete. In this paper, a path-edge matrix for the graph G is presented by which it is possible to classify the graphs in which η(G)=2. It is also proved that there is no graph G with the property that η(G)=1 or η(G)=3. Finally, it is proved that, for a given positive integer k,k≠1,3, there exists a graph G with η(G)=k.

On the revised Szeged index

We give bounds for the revised Szeged index, and determine the n -vertex unicyclic graphs with the smallest, the second-smallest and the third-smallest revised Szeged indices for n ≥ 5 , and the n -vertex unicyclic graphs with the k th-largest revised Szeged indices for all k up to 3 for n = 5 , to 5 for n = 6 , to 6 for n = 7 , to 7 for n = 8 , and to ⌊ n 2 ⌋ + 4 for n ≥ 9 . We also determine the n -vertex unicyclic graphs of cycle length r , 3 ≤ r ≤ n , with the smallest and the largest revised Szeged indices.

An Exact Expression for the Szeged Index of V-Phenylenic Nanotubes and Nanotori

An Exact Expression for the Szeged Index of V-Phenylenic Nanotubes and Nanotori

Use of the Szeged index and the revised Szeged index for measuring network bipartivity

We have revisited the Szeged index ( S z ) and the revised Szeged index ( S z ∗ ) , both of which represent a generalization of the Wiener number to cyclic structures. Unexpectedly we found that the quotient of the two indices offers a novel measure for characterization of the degree of bipartivity of networks, that is, offers a measure of the departure of a network, or a graph, from bipartite networks or bipartite graphs, respectively. This is because the two indices assume the same values for bipartite graphs and different values for non-bipartite graphs. We have proposed therefore the quotient S z / S z ∗ as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient S z / S z ∗ illustrated on a number of smaller graphs as models of networks.

SOME TOPOLOGICAL INDICES OF NANOSTAR DENDRIMERS

Wiener index is a topological index based on distance between every pair of vertices in a graph G. It was introduced in 1947 by one of the pioneer of this area e.g, Harold Wiener. In the present paper, by using a new method introduced by klavžar we compute the Wiener and Szeged indices of some nanostar dendrimers.

Szeged index, edge Szeged index, and semi-star trees

A semi-star tree is a star tree whose some edges may be replaced by paths of length more than one. In this paper we present some increasing and decreasing transformations for Szeged index of the semi-star trees. Then we introduce palm semi-star tree and uniform semi-star tree and show that they are extremal with respect to the Szeged index and edge Szeged index. In addition, we investigate the relation between the Szeged index and edge Szeged index for all trees.

Topological Relationship Between Wiener, Padmaker‐Ivan, and Szeged Indices and Energy and Electric Moments in Armchair Polyhex Nanotubes with the Same Circumference and Varying Lengths

Carbon nanotubes are carbon nanostructure compounds that have been synthesized and identified in an attractive array of structures and forms. Graph theory is a branch of mathematics related to topology and combinatorial problems. Chemical graph theory has been extensively applied to predicting the physical properties of small molecules through quantitative structure‐property relationships (QSPRs). A large number of topographical indices (TIs) have been reported, and strong correlations have been demonstrated between many physical properties and one or more topological indices. In this study, relationships are presented between topological indices such as the Wiener, Padmakar‐Ivan and Szeged indices, and the electric moments and energies (kJmol−1) of some armchair polyhex carbon nanotubes TUVC6[2p,q] with a fixed circumference [2p] and various lengths [q]. The results are performed by Becke's 3‐parameter formulation with the Lee‐Yang‐Parr functional (B3LYP) method on the 3‐21G basis set.

Maximum values of Szeged index and edge-Szeged index of graphs

AbstractThe Szeged index is a graph invariant which is a natural generalization of Wienerindex. In this note, we disprove two recent conjectures concerning with the maxi-mum value of Szeged index of graphs, which are due to Khalifeh et al. (Europ. J.Combinatorics (2008), doi:10.1016/j.ejc.2008.09.019) and respectively, to Gutmanet al. (Groat. Chem. Acta 81(2)(2008) 263-266) and prove a conjecture on Szegedindex due to Klavzar et al. ( Appl. Math. Lett. 9(1996), 45-49), which states thatthe complete bipartite graph K n2 , n2 has maximum Szeged index among all con-nected graphs on n vertices. The last conjecture is previously proved by Dobrynin(Croat. Chem. Acta 70(3), 819-825), but our proof turns out to be much simplerand self-contained.Keywords: Szeged index, Wiener index 1 Introduction All graphs considered here are finite, undirected and simple. We refer to [4] forunexplained terminology and notations. Let G =(V(G),E(G)) be a graph. 1 Supported by Canada Research Chair Program Electronic Notes in Discrete Mathematics 34 (2009) 405–4091571-0653/$ – see front matter Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.www.elsevier.com/locate/endmdoi:10.1016/j.endm.2009.07.067

Estimating the Szeged index

Lower and upper bounds on Szeged index of connected (molecular) graphs are established as well as Nordhaus–Gaddum-type results, relating the Szeged index of a graph and of its complement.

Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑{x,y}⊆Vd(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑e=uv∈Enu(e|G)nv(e|G), where nu(e|G) (resp. nv(e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑e=uv∈E[neu(e|G)+nev(e|G)], where neu(e|G) (resp. nev(e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.

Edge-contributions of some topological indices and arboreality of molecular graphs

Some graph invariants can be computed by summing certain values, called edge-contributions over all edges of graphs. In this note we use edge-contributions to study relationships among three graph invariants, also known as topological indices in mathematical chemistry: Wiener index, Szeged index and recently introduced revised Szeged index. We also use the quotient between the Wiener index and the revised Szeged index to study tree-likeness of graphs.

Some new results on distance-based graph invariants

We study distance-based graph invariants, such as the Wiener index, the Szeged index, and variants of these two. Relations between the various indices for trees are provided as well as formulas for line graphs and product graphs. This allows us, for instance, to establish formulas for the edge Wiener index of Hamming graphs, C 4 -nanotubes and C 4 -nanotori. We also determine minimum and maximum of certain indices over the set of all graphs with a given number of vertices or edges. Finally, we study the order of magnitude of the edge Wiener and edge Szeged index, responding negatively to a conjecture that is related to the maximization of the edge Szeged index.

Szeged index of [formula omitted] nanotubes

The Szeged index of a graph G is defined as S z ( G ) = ∑ e ∈ E ( G ) n u ( e ) n v ( e ) , where n u ( e ) is the number of vertices of G lying closer to u than to v , n v ( e ) is the number of vertices of G lying closer to v than to u and the summation goes over all edges e = u v of G . In this paper we find an exact expression for Szeged index of T U C 4 C 8 ( S ) nanotubes, using a theorem of A. Dobrynin and I. Gutman on connected bipartite graphs (see [A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math. Nouvelle ser. tome 56 (70) (1994) 18–22]).

Vertex-PI Index of Some Nanotubes

The vertex version of PIindex is a molecular structure de- scriptor which is similar to vertex version of Szeged index. In this pa- per, we compute the vertex-PIindex of TUC4C8(S), TUC4C8(R) and HAC5C7(r, p).

Relationship Between PI and Szeged Indices of a Triangulane and Its Associated Dendrimer

Relationship Between PI and Szeged Indices of a Triangulane and Its Associated Dendrimer

A matrix method for computing Szeged and vertex PI indices of join and composition of graphs

The Szeged index extends the Wiener index for cyclic graphs by counting the number of vertices on both sides of each edge and sum these counts. Klavzar et al. [S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9 (5) (1996) 45–49] provided an exact formula for computing Szeged index of product of graphs. In this paper, we apply a matrix method to obtain exact formulae for computing the Szeged index of join and composition of graphs. The join and composition of the vertex PI index of graphs are also computed.

Szeged index of some nanotubes

Abstract The Szeged index is one of the most important topological indices defined in chemistry. In this paper, the Szeged index of the hexagonal triangle graph T(n) and the zig-zag polyhex nanotube TUHC6[2p,q] are computed.

PI and Szeged Indices of One-Pentagonal Carbon Nanocones

Carbon nanocones originally discovered by Ge and Sattler in 1994.1 These are constructed from a graphene sheet by removing a 60 wedge and joining the edges produces a cone with a single pentagonal defect at the apex, Figure 1. Removing additional wedges introduces more such defects and reduces the opening angle. A cone with six pentagons has an opening angle of zero and is just a nanotube with one open end. We now recall some algebraic definitions that will be used in the paper. Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V (G) and E(G), respectively. A topological index of a graph G is a numeric quantity related to G. The oldest topological index is the Wiener index which introduced by Harold Wiener.2 Khadikar and co-authors3–7 defined a new topological index and named it Padmakar-Ivan index. They abbreviated this new topological index as PI. This newly proposed topological index does not coincide with the Wiener index for acyclic molecules. It is defined as PI(G) = ∑ e∈G neu e G +nev e G)], where neu e G) is the number of edges of G lying closer to u than to v and nev e G) is the number of edges of G lying closer to v than to u. Edges equidistant from both ends of the edge uv are not counted. The Szeged index is another topological index which is introduced by Ivan Gutman.8–10 To define the Szeged index of a graph G, we assume that e = uv is an edge connecting the vertices u and v. Suppose Meu e G) is the number of vertices of G lying closer to u and Mev e G) is the number of vertices of G lying closer to v. Then the Szeged index of the graph G is defined as Sz(G) = ∑ e=uv∈E G Meu e G)Mev e G). Notice that vertices equidistance from u and v are not taken into account.

Computing the Szeged index of third and fourth dendrimer nanostars

Let e be an edge of a graph, G, connecting the vertices u and v. Define two sets N1(e|G) and N2(e|G) as N1(e|G)={x∈V(G)|d(x, u)<d(x, v)} and N2(e|G)={x∈V(G)|d(x, v)<d(x, u)}. The number of elements of N1(e|G) and N2(e|G) are denoted by n1(e|G) and n2(e|G), respectively. The Szeged index of the graph G is defined as Sz(G)=∑e∈E(G)n1(e|G) n2(e|G). In this Letter, the Szeged index of third and fourth dendrimer nanostars is computed.

A simple method for simultaneous estimation of Wiener (W) and Szeged (Sz) indices of benzenoids

This paper describes a simple method for the simultaneous estimation of Wiener (W) and Szeged (Sz) indices for benzenoid systems using elementary cut method. The method is extended to cyclic graphs containing acyclic (tree-like) side chains.