Szeged Index Research Articles

Cut Method: Update on Recent Developments and Equivalence of Independent Approaches

The cut method is a powerful tool for the investigation of distance-based (and some other) molecular structure-descriptors. In this paper a survey on the recent developments of the method is given. The instances of the standard cut method for the Wiener index, the Szeged index, the PI index, the generalized terminal Wiener index, the Gutman index, the edge-Wiener index, and the edge-Szeged index are described, where a standard cut method is a method that applies to partial cubes. It is pointed out that the standard cut method was recently independently discovered a couple of times. Numerous proper extensions of the standard cut method are presented. The method extends to l1-graphs, graphs with a non-trivial canonical metric representation, graphs with transitive relation Θ, and partial Hamming graphs. The instances of these extended cut methods include the Wiener index, the degree distance, distance moments, and the colored Wiener index. Keywords: Colored Wiener index, cut method, distance moment, isometric embedding, partial cube, PI index, QSPR/QSAR, szeged index, terminal wiener index, topological index, wiener index.

Relationship between Topological Indices and Thermodynamic Properties and of the Monocarboxylic Acids Applications in QSPR

Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. One of the useful indices for examination of structure- property relationship is Randic' index. In this study is represented the relationship between the Randic', Balaban and Szeged indices and Harary numbers to the enthalpies of combustion (  C H sub H  ) of monocarboxylic acids (C2C20) are established, and then, some useful topological indices for examination of the structure- property relationship are presented.

Comparison between the Szeged index and the eccentric connectivity index

Let Sz(G) and ξc(G) be the Szeged index and the eccentric connectivity index of a graph G, respectively. In this paper we obtain a lower bound on Sz(T)−ξc(T) by double counting on some matrix and characterize the extremal graphs. From this result we compare the Szeged index and the eccentricity connectivity index of trees. For bipartite graphs we also compare the Szeged index and the eccentricity connectivity index. Moreover, we show that Sz(G)−ξc(G)≥−4 for bipartite graphs and this result is not true in the general case. Finally, we classify the bipartite graphs G in which Sz(G)−ξc(G)∈{−4,−3,−2,−1,0,1,2}.

A congruence relation for Wiener and Szeged indices

In a recent paper [H. Lin, MATCH Communications in Mathematical and in Computer Chemistry 70 (2013) 575-582], a congruence relation for Wiener indices of a class of trees was reported. We now show that Lin?s congruence is a special case of a much more general result.

STUDYING THE CORONA PRODUCT OF GRAPHS UNDER SOME GRAPH INVARIANTS

‎‎The corona product $Gcirc H$ of two graphs $G$ and $H$ is‎ ‎obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$;‎ ‎and by joining each vertex of the $i$-th copy of $H$ to the‎ ‎$i$-th vertex of $G$‎, ‎where $1 leq i leq |V(G)|$‎. ‎In this‎ ‎paper‎, ‎exact formulas for the eccentric distance sum and the edge‎ ‎revised Szeged indices of the corona product of graphs are‎ ‎presented‎. ‎We also study the conditions under which the corona‎ ‎product of graphs produces a median graph‎.

On the difference between the revised Szeged index and the Wiener index

Let Sz⋆(G) and W(G) be the revised Szeged index and the Wiener index of a graph G. Chen et al. (2014) proved that if G is a non-bipartite connected graph of order n≥4, then Sz⋆(G)−W(G)≥(n2+4n−6)/4. Using a matrix method we prove that if G is a non-bipartite graph of order n, size m, and girth g, then Sz⋆(G)−W(G)≥n(m−3n4)+P(g), where P is a fixed cubic polynomial. Graphs that attain the equality are also described. If in addition g≥5, then Sz⋆(G)−W(G)≥n(m−3n4)+(n−g)(g−3)+P(g). These results extend the bound of Chen, Li, and Liu as soon as m≥n+1 or g≥5. The remaining cases are treated separately.

Tricyclic graphs with maximal revised Szeged index

The revised Szeged index of a graph G is defined as Sz∗(G)=∑e=uv∈E(nu(e)+n0(e)/2)(nv(e)+n0(e)/2), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n0(e) is the number of vertices equidistant to u and v. In this paper, we give an upper bound of the revised Szeged index for a connected tricyclic graph, and also characterize those graphs that achieve the upper bound.

A New Technique to Compute Padmakar-Ivan Index and Szeged Index of Pericondensed Benzenoid Graphs

A New Technique to Compute Padmakar-Ivan Index and Szeged Index of Pericondensed Benzenoid Graphs

Revised Szeged Index and Revised Edge Szeged Index of Certain Special Molecular Graphs

In theoretical chemistry, the revised Szeged index and revised Szeged edge index were introduced to measure the stability of alkanes and the strain energy of cycloalkanes. In this paper, by virtue of mathematical derivation, we determine the revised Szeged index and revised edge Szeged index of fan molecular graph, wheel molecular graph, gear fan molecular graph, gear wheel molecular graph, and their r-corona molecular graphs. These molecular structures are widely used in biology, medical science and pharmaceutical fields. At last, the normalized revised Szeged indexes of fan molecular graph, wheel molecular graph, gear fan molecular graph, gear wheel molecular graph, and their r-corona molecular graphs are given.

The (revised) Szeged index and the Wiener index of a nonbipartite graph

Hansen et al. used the computer program AutoGraphiX to study the differences between the Szeged index Sz(G) and the Wiener index W(G), and between the revised Szeged index Sz∗(G) and the Wiener index for a connected graph G. They conjectured that for a connected nonbipartite graph G with n≥5 vertices and girth g≥5, Sz(G)−W(G)≥2n−5, and moreover, the bound is best possible when the graph is composed of a cycle C5 on 5 vertices and a tree T on n−4 vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph G with n≥4 vertices, Sz∗(G)−W(G)≥n2+4n−64, and moreover, the bound is best possible when the graph is composed of a cycle C3 on 3 vertices and a tree T on n−2 vertices sharing a single vertex. In this paper, we not only give confirmative proofs to these two conjectures but also characterize those graphs that achieve the two lower bounds.

Bicyclic graphs with maximal revised Szeged index

The revised Szeged index of a graph G is defined as Sz∗(G)=∑e=uv∈E(nu(e)+n0(e)/2)(nv(e)+n0(e)/2), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n0(e) is the number of vertices equidistant to u and v. Hansen et al. used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph G of order n≥6: Sz∗(G)≤{(n3+n2−n−1)/4,if n is odd ,(n3+n2−n)/4,if n is even . with equality if and only if G is the graph obtained from the cycle Cn−1 by duplicating a single vertex. This paper is to give a confirmative proof to this conjecture.

Revised and edge revised Szeged indices of graphs

The revised Szeged index is a molecular structure descriptor equal to the sum of products [ n u ( e ) + n 0 ( e ) / 2] × [ n v ( e ) + n 0 ( e ) / 2] over all edges e = u v of the molecular graph G , where n 0 ( e ) is the number of vertices equidistant from u and v , n u ( e ) is the number of vertices whose distance to vertex u is smaller than the distance to vertex v and n v ( e ) is defined analogously. In this paper, new formula for computing this molecular descriptor is presented by which it is possible to reprove most of results given in [M. Aouchiche and P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010), 1662–1666]. We also present an edge version of this graph invariant. At the end of the paper an open question is presented.

The Topological Study of an Infinite Family of Fullerenes with 10n Carbon Atoms

Fullerenes are carbon cage molecules having 12 pentagonal and (n/2 – 10) hexagonal faces, where 20 ≤ n (≠ 22) is an even integer. In this paper, an infinite class of fullerenes with 10n carbon atoms is investigated. We prove that the vertex–PI, Szeged and revised Szeged indices of this family are computed by formulas 150n2 − 100n, 250n3 + 3075n − 13800 and 250n3 + 250n2 + 4275n − 15650, respectively, when n > 10 is a positive integer. A MATLAB program is also presented that is useful for our calculations.

Wiener index versus Szeged index in networks

Let (G,w) be a network, that is, a graph G=(V(G),E(G)) together with the weight function w:E(G)→R+. The Szeged index Sz(G,w) of the network (G,w) is introduced and proved that Sz(G,w)≥W(G,w) holds for any connected network where W(G,w) is the Wiener index of (G,w). Moreover, equality holds if and only if (G,w) is a block network in which w is constant on each of its blocks. Analogous result holds for vertex-weighted graphs as well.

On topological indices for small RNA graphs

The secondary structure of RNAs can be represented by graphs at various resolutions. While it was shown that RNA secondary structures can be represented by coarse grain tree-graphs and meaningful topological indices can be used to distinguish between various structures, small RNAs are needed to be represented by full graphs. No meaningful topological index has yet been suggested for the analysis of such type of RNA graphs. Recalling that the second eigenvalue of the Laplacian matrix can be used to track topological changes in the case of coarse grain tree-graphs, it is plausible to assume that a topological index such as the Wiener index that represents all Laplacian eigenvalues may provide a similar guide for full graphs. However, by its original definition, the Wiener index was defined for acyclic graphs. Nevertheless, similarly to cyclic chemical graphs, small RNA graphs can be analyzed using elementary cuts, which enables the calculation of topological indices for small RNAs in an intuitive way. We show how to calculate a structural descriptor that is suitable for cyclic graphs, the Szeged index, for small RNA graphs by elementary cuts. We discuss potential uses of such a procedure that considers all eigenvalues of the associated Laplacian matrices to quantify the topology of small RNA graphs.

THE TOPOLOGICAL STUDY OF IPR FULLERENES BY SZEGED AND REVISED SZEGED INDICES

In this paper the Szeged and revised Szeged indices of an infinite family of IPR fullerenes with 50 + 10n carbon atoms and D 5h point group symmetry are computed. It seems that some chemico-physical properties of these fullerenes are derived from the mentioned topological invariants.

Eccentric-connectivity, GA, Harary, hyper-Wiener and Wiener Polarity indices of Helicenes

In this paper we compute eccentric-connectivity index, GA index, Harary index, Hyper -wiener index and Wiener polarity index of Helicines. A comparative study of these indices along with the known topological indices Wiener index, Szeged index, PI index, Zagreb index, Sh values is made using their log P values.

COMPUTING THE SZEGED INDEX OF 4,4׳-BIPYRIDINIUM DENDRIMER

Let e be an edge of a 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) ( ) ( ) 1 2 n e G n e G e E    . In this paper we compute the Szeged index of a 4,4 ׳-Bipyridinium dendrimer.

PI, SZEGED AND REVISED SZEGED INDICES OF IPR FULLERENES

In this paper PI, Szeged and revised Szeged indices of an infinite family of IPR fullerenes with exactly 60+12n carbon atoms are computed. A GAP program is also presented that is useful for our calculations.

Theoretical Study of the Relationship Between Wiener, Padmakar-Ivan, and Szeged Topological Indices in Contrast to Energy, Electric Moments and Partition Coefficient of Armchair Polyhex Carbon Nanotubes with Various Circumference and Fixed Lengths

The use of theoretical tools for the molecular modeling of physicochemical processes and chemical reaction is one of the main distinctive characteristics of modern chemistry. Topological indices have attracted more attention from theoretical chemists and others and have become the focus of computational chemistry. Chemical graph theory is a branch of mathematics which combines graph theory and chemistry and has been extensively applied to predicting the physical properties of small molecules through Quantitative Structure Property Relationship (QSPR). Quantum chemical calculations are thus an attractive source of new molecular descriptors, which can, in principle, express all of the electronic and geometric properties of molecules and their intractions. Electric moments and energies of molecules are examples of quantum chemical descriptors used in QSPR studies. In this study, the relationship between some of the topological indices such as Wiener, Padmakar-Ivan, Szeged indices and partition coefficient (Log P) in contrast to the electric moments and energy (kJmol−1) of some armchair polyhex carbon nanotubes TUVC6[2p,q] with various circumference and fixed lengths are represented. Each nanotube is optimized at the level of the Becke3, Lee-Yang-Parr (B3LYP) method, using the 3–21G standard basis set. The relationship between topological and quantum chemical descriptors are discussed in this study.