Discrete Applied Mathematics (0166218X)307pp. 172-179
The exponential second Zagreb index of a connected graph G, denoted by eM(G), is defined as the sum of the weights ed over all edges uv∈E(G), where dG(u) denotes the degree of vertex u. The maximum value of eM in the class of connected acyclic graphs has already been studied. In this paper, maximal unicyclic and bicyclic graphs of order n with respect to eM are represented. © 2021 Elsevier B.V.
Journal of Applied Mathematics and Computing (15985865)59(1-2)pp. 37-46
The first degree-based entropy of a connected graph G is defined as: I1(G)=log(∑vi∈V(G)deg(vi))-∑vj∈V(G)deg(vj)logdeg(vj)∑vi∈V(G)deg(vi). In this paper, we apply majorization technique to extend some known results about the maximum and minimum values of the first degree-based entropy of trees, unicyclic and bicyclic graphs. © 2018, Korean Society for Computational and Applied Mathematics.
Linear and Multilinear Algebra (03081087)67(9)pp. 1736-1749
Let G be a simple and undirected graph with Laplacian polynomial ψ(G, λ) =∑nk=0 (− 1)n−k ck (G)λk. In earlier works, some formulas for computing c2(G), cn-2(G) and cn−3(G) in terms of the number of vertices, the Wiener, the first Zagreb and the forgotten indices are given. In this paper, we continue this work by computing cn−4(T), where T is a tree. A lower and an upper bound for cn−4(T) are obtained. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
Utilitas Mathematica (03153681)111pp. 189-197
Ashrafi a.r., A.R.,
Eliasi, M.,
Ghalavand a., A.,
Ori, O. Acta et Commentationes Universitatis Tartuensis de Mathematica (14062283)22(2)pp. 261-278
Let G be an n-vertex graph with the vertex degree sequence d1; d2, …, dn. The Narumi-Katayama index of G is defined as NK(G) = ∏n i=1di. We determine eight classes of n-vertex tricyclic graphs with the fit through the eighth maximal NK index, n ≥ 10. We also identify ten classes of n-vertex tetracyclic graphs with the fit through the ninth maximal NK index, n ≥ 10, and thirteen classes of n-vertex pentacyclic graphs with the fit through the twelfth maximal NK index, n ≥ 12. © 2018, University of Tartu. All right reserved.
Match (03406253)79(3)pp. 645-657
Determining extremal values of graph entropies for some given classes of graphs is intricate, because there is a lack of analytical methods to tackle this particular problem. In this paper we apply the strong mixing variables method for this propose. We characterized the graphs which attain the minimum values of the graph entropy, based on an arbitrary increasing convex information functional, among certain classes of graphs, namely, trees, unicyclic graphs and bicyclic graphs. © 2018 University of Kragujevac, Faculty of Science. All rights reserved.
Kragujevac Journal of Mathematics (14509628)42(3)pp. 325-333
For simple graph G with edge set E(G), the second Zagreb index of G is defined as M2(G) = Σuv∈E(G)[dG(u)dG(v)], where dG(v) is the degree of the vertex v in G. In this paper, we identify the nine classes of trees, which have the first to the sixth smallest second Zagreb indices, among all the trees of the order n ≥ 11. © University of Kragujevac - Faculty of Science, 2018.
Ars Combinatoria (03817032)117pp. 147-153
The Szeged polynomial of a connected graph G, is defined as Sz(G,x) = σ eεE(G)xnu (e)nv(e) where nu(e) is the number of vertices of G lying closer to u than to ν, nu(e) is the number of vertices of G lying closer to ν than to u and the summation goes over all edges e = uν ε E(G) of G. Ashrafi et. al. (On Szeged polynomial of a graph, Bui. Iran. Math. Soc. 33 (2007) 37-46.) proved that if the number of the vertices of G is even, then deg(Sz(G, x)) < 1/4 [V(G)2] where V(G) is the set of vertices of G. In this paper we study the structure of graphs, with even number of vertices, for which the equality holds. Also we examine equality for the sum of graphs. Copyright © 2014, Charles Babbage Research Centre.
Match (03406253)69(3)pp. 765-773
Let G = (V, E) be a simple graph with n = |V | vertices and m = |E| edges. The first and the second Zagreb indices of G are defined as M1(G) = Σ uεV du2 = Σ uvεE[du + dv] and M2(G) = Σ uvεE du dv, respectively, where du denotes the degree of vertex u. We compare the multiplicative versions of these indices.
Match (03406253)69(1)pp. 111-119
The Hosoya polynomial of a graph G is a graphical invariant polynomial that its first derivative at x = 1 is equal to the Wiener index. In this paper we compute the Hosoya polynomial of the hierarchical product of graphs and give some applications of this operation.
Current Nanoscience (18756786)9(4)pp. 502-513
A comparison of the performance of some specific topological indices was conducted. The study was concentrated on determining which topological indices are included in the best linear correlation models constructed for modeling the total energy of zig-zag nanotubes. The normalization of some particular widely used topological indices such as the Wiener index, Shultz index, hyper Wiener index, Harary index, Szeged index, multiplicative Wiener index and reciprocal complementary-Wiener index plus total energy of 56 zigzag polyhex nanotubes was considered in this study. © 2013 Bentham Science Publishers.
Match (03406253)68(1)pp. 217-230
The first Zagreb index of a graph G, with vertex set V(G) and edge set E(G), is defined as M1(G) = Σuε∈V(G)d(u) 2 where d(u) denotes the degree of the vertex ε. An alternative expression for M1(G) is Σu∈εE(G)[d(u) + d(ε)]. We consider a multiplicative version of M1 defined as Π1(G) = Πuε∈E(G)[d(u) + d(v)]. We prove that among all connected graphs with a given number of vertices, the path has minimal Π1. We also determine the trees with the second-minimal Π1.
Current Nanoscience (18756786)8(4)pp. 566-570
By a method that already has been introduced by Eliasi and Taeri, we computed the diameter of zigzag polyhex nanotubes. As a consequence of calculating, the reverse Wiener index and reciprocal complementary Wiener index of zigzag polyhex nanotorus were computed. We expressed the results in terms of Gamma function, to help us use the software to speed up calculations. Finally, we obtain these indices for some numerical values. © 2012 Bentham Science Publishers.
Discrete Applied Mathematics (0166218X)160(9)pp. 1333-1344
We determine the Wiener index of graphs which are constructed by some operations such as Mycielski's construction, generalized hierarchical product and t-th subdivision of graphs. © 2012 Elsevier B.V. All rights reserved.
Utilitas Mathematica (03153681)84pp. 105-117
The multiplicative Wiener index, π (G) , is equal to the product of distances between all pairs of vertexes of a (molecular) graph G. In this paper we compute this index for some nanotubes and nanotori by consider them as cartesian product of paths and cycles. Also we compute this index for some composite graphs.
Ars Combinatoria (03817032)99pp. 279-287
Let G be a non-abelian group and let Z(G) be the center of G. Associate with G a graph ΓG as follows: Take G\Z(G) as vertices of ΓG and join two distinct vertices x and y whenever xy ≠ yx. Graph ΓG is called the non-commuting graph of G and many of graph theoretical properties of ΓG have been studied. In this paper we study some metric graph properties of ΓG.
Applied Mathematics Letters (18735452)24(4)pp. 582-587
The ordinary generalized geometricarithmetic index of graphs is introduced and some properties especially lower and upper bounds in terms of other graph invariants and topological indices are obtained. © 2010 Elsevier Ltd. All rights reserved.
Discrete Applied Mathematics (0166218X)159(8)pp. 866-871
The hyper Wiener index of the connected graph G is WW(G)=12∑u, v⊆V(G)(d(u,v)+d(u,v)2), where d(u,v) is the distance between the vertices u and v of G. In this paper we compute the hyper-Wiener index of the generalized hierarchical product of two graphs and give some applications of this operation. © 2011 Elsevier B.V. All rights reserved.
Optoelectronics and Advanced Materials, Rapid Communications (18426573)4(4)pp. 565-567
The multiplicative Wiener index, π (G), is equal to the product of the distances between all pairs of vertices of the underlying molecular graph G. In this paper we compute this index for zigzag polyhex nanotubes.
Physica A: Statistical Mechanics and its Applications (03784371)389(14)pp. 2733-2738
The quantum vibrational partition function has been obtained in the Tsallis statistics framework for the entropic index, q, between 1 and 2. The effect of non-extensivity on the population of states and thermodynamic properties have been studied and compared with their corresponding values obtained in the Boltzmann-Gibbs (BG) statistics. Our results show that the non-extensive partition function of harmonic oscillator at any temperature is larger than its corresponding values for an extensive system and that their differences increase with temperature and entropic index. Also, the number of accessible states increases with q but, compared to the BG statistics, the occupation number decreases for low energy levels while the population of the higher energy levels increases. The internal energy and heat capacity have also been obtained for the non-extensive harmonic oscillator system. Results indicate that the heat capacity is greater than its corresponding value in the extensive (BG) system at low temperatures but that this trend is reversed at higher temperatures. © 2010 Elsevier B.V. All rights reserved.
Current Nanoscience (18756786)5(4)pp. 514-518
Chemical compounds are often modeled as polygonal shapes, where a vertex represents an atom and an edge symbolizes a bond. Topological properties of molecular graphs of chemical compounds can be correlated to their chemical properties and biological activities. Topological indices are the oldest and the most widely used to describing these activity relationships. Many topological indices can be expressed in terms of the distance concept in graphs. In this paper we explain a method, using the concept of distance in the graphs of zigzag polyhex nanotubes, which enables us to compute different topological indices simultaneously. © 2009 Bentham Science Publishers Ltd.
Discrete Applied Mathematics (0166218X)157(4)pp. 794-803
The Wiener index is the sum of distances between all vertex pairs in a connected graph. This notion was motivated by various mathematical properties and chemical applications. In this paper we introduce four new operations on graphs and study the Wiener indices of the resulting graphs. © 2008 Elsevier B.V. All rights reserved.
Digest Journal of Nanomaterials and Biostructures (18423582)4(4)pp. 757-762
The Harary index, H = H(G), of a molecular graph G is based on the concept of reciprocal distance and is defined, in parallel to the Wiener index, as the half-sum of the off-diagonal elements of the molecular distance matrix of G. In this paper we compute the Harary index of zigzag polyhex nanotorus.
ANZIAM Journal (14461811)50(1)pp. 75-86
The hyper-Wiener index of a connected graph G is defined as $WW(G)=(1/4)∑ (u,v) V(G)× V(G) (d(u,v)+d(u,v) ), where V (G) is the set of all vertices of G and d(u,v) is the distance between the vertices u,vV (G). In this paper we find an exact expression for the hyper-Wiener index of TUHC6[2p,q], the zigzag polyhex nanotube. Copyright © Australian Mathematical Society 2009.
Asian Journal of Chemistry (09707077)21(2)pp. 931-941
The Schultz polynomial, S(G,x), of a molecular graph G has the property that its first derivative at x=l is equal to the Schultz index of graph. Ivan Gutman discovered that in the case of G is a tree, S(G,x), has closely related to the Wiener polynomial of G. In this paper, we find the exact expression for Schultz polynomial of TUHC6 [2p; q], the zigzag polyhex nanotubes, and obtain a relation between Schultz and Wiener polynomials of TUHC6 [2p; q].
Journal of Theoretical and Computational Chemistry (17936888)7(5)pp. 1029-1039
Graph theory was successfully applied in developing a relationship between chemical structure and biological activity. The concept of distance in graphs is basic in the definition of various topological indices for chemical compounds, which determines some of the physicochemical properties of them. In this paper, we explain a method, using the concept of distance in the graph of zigzag polyhex nanotorus, which enables us to compute different topological indices simultaneously. © 2008 World Scientific Publishing Company.
Applied Mathematics Letters (18735452)21(9)pp. 916-921
The Wiener index W (G) of a connected graph G is defined as the sum of distances between all pairs of vertices. The Wiener polynomial H (G, x) has the property that its first derivative evaluated at x = 1 equals the Wiener index, i.e. H′ (G, 1) = W (G). The hyper-Wiener polynomial H H (G, x) satisfies the condition H H′ (G, 1) = W W (G), the hyper-Wiener index of G. In this paper we introduce a new generalization W (G, y) of the Wiener index and H (G, x, y) of the Wiener polynomial. One of the advantages of our definitions is that one can handle the Wiener and hyper-Wiener index (respectively polynomial) with the same formula, i.e. W (G) = W (G, 1), W W (G) = W (G, 2), H (G, x) = H (G, x, 1) and H H (G, x) = H (G, x, 2). © 2007 Elsevier Ltd. All rights reserved.
Journal of the Serbian Chemical Society (03525139)73(3)pp. 311-319
The Hosoya polynomial of a molecular graph G is defined as H(G,λ) = ∑{u, v}⊆V(G)λd(u,v), where d(u,v) is the distance between vertices u and v. The first derivative of H(G,λ) at λ = 1 is equal to the Wiener index of G, defined as W(G) = ∑ {u, v}⊆V(G)d(u,v). The second derivative of 1/2λH(G, λ) at λ = 1 is equal to the hyper-Wiener index, defined as WW(G) = 1/2W(G)+1/2∑{u, v}⊆V(G)d(u,v)2. Xu et al. 1 computed the Hosoya polynomial of zigzag open-ended nanotubes. Also Xu and Zhang2 computed the Hosoya polynomial of armchair open-ended nanotubes. In this paper, a new method was implemented to find the Hosoya polynomial of G = HC6[p,q], the zigzag polyhex nanotori and to calculate the Wiener and hyper Wiener indices of G using H(G,λ).
International Journal Of Molecular Sciences (14220067)9(10)pp. 2016-2026
The study of topological indices – graph invariants that can be used for describing and predicting physicochemical or pharmacological properties of organic compounds – is currently one of the most active research fields in chemical graph theory. In this paper we study the Schultz index and find a relation with the Wiener index of the armchair polyhex nanotubes T UV C6[2p, q]. An exact expression for Schultz index of this molecule is also found. © 2008 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland.
Applicable Analysis and Discrete Mathematics (14528630)2(2)pp. 285-296
For a connected graph G, the Schultz and modified Schultz polynomials, introduced by I. Gutman: Some relations between distance-based polynomials of trees. Bulletin, Classe des Sciences Math́ematiques et Naturelles, Sciences math́ematiques, Vol. CXXXI, 30 (2005) 1-7, are defined as H1(G,x) = 1/2∑{(δu + δv)xd(u,v,|G) | u,v,∈ V(G), u ≠ v} and H2(G,x) = 1/2∑{(δuδv)xd(u,v,|G) | u,v,∈ V(G), u ≠ v}, respectively, where δu is the degree of vertex u, d(u, v|G) is the distance between u and v and V(G) is the vertex set of G. In this paper we find identities for the Schultz and modified Schultz polynomials of the sum, join and composition of graphs. As an application of our results we find the Schultz polynomial of C4 nanotubes.
Match (03406253)59(2)pp. 437-450
Topological indices of nanotubes are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. The Szeged index is obtained as a bond additive quantity where bond contributions are given as the product of the number of atoms closer to each of the two end points of each bond. In this paper we find an exact expression for Szeged index of TUVC6[2p, q], the armchair polyhex nanotubes, using a theorem of A. Dobrynin and I. Gutman on connected bipartite graphs (see Ref [1]).
Journal of Computational and Theoretical Nanoscience (15461955)4(6)pp. 1174-1178
Topological indices of nanotubes are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. The Balaban index of a molecular graph calculates the average distance sum connectivity index. Balaban index measures the ramification and it tends to increase with molecular ramification. In this paper we derive the exact expressions for Balaban index of zigzag polyhex nanotorus. Copyright © 2007 American Scientific Publishers. All rights reserved.
Ars Combinatoria (03817032)85pp. 307-318
The hyper Wiener index of a connected graph G is defined as WW(G) = 1/2 Σ{u,v}⊆V(G) d(u, v)+1/2 Σ{u, v}1/2V(G) d(u, v) where d(u, v) is the distance between vertices u, v E V(G). In this paper we find an exact expression for hyper Wiener index of HC6[p, q], the zigzag polyhex nanotori.
AIP Conference Proceedings (0094243X)929pp. 243-249
Topological indices of nanotube? are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. Harold Wiener in 1947 introduced the notion of path number of a graph as the sum of the distances between two carbon atoms in the molecules, in terms of carbon-carbon bound. The Wiener index of graph G is defined as W(G)= 1/2 ∑u,v∈V(G) d(u,v), where V(G) is the set of vertices of the graph and d(u,v) is the distance between two vertices u,v. The hyper Wiener index of G is defined by WW(G)= 1/2 W(G) +1/4 ∑uv∈V(G) d(u,v)2. In this paper we present some new results on topological indices of nanotubes and calculate hyper Wiener index of some nanotubes. © 2007 American Institute of Physics.
Utilitas Mathematica (03153681)74pp. 55-64
Schultz index of a molecular graph G is defined as 1/2 ∑ {u,v}⊂V(G)(deg (u) + deg (v))d(u,v), where d(u,v) is the distances between u and v in the graph G and deg (u) is the degree of the vertex u. In this paper we find an exact expression for Schultz index of TUHC6[2p,q], the zigzag polyhex nanotubes.
Match (03406253)56(2)pp. 383-402
The Sezegd index of a graph G is defined as Sz(G)= Σ eεE(G) nu(e)nv(e), where nu(e) is the number of vertices of G lying closer to u than to v, nv(e) is the number of vertices of G lying closer to v than to u and the summation goes over all edges e = uv of G. Also Balaban index of G is defined by J(G) = m/(μ + 1) ΣuvεE(G) [d(u)d(v)]-0.5, where d(v) = ΣxεV(G) d(v, x), is the summation of distances between v and all vertices of G, m is the number of edges in G and μ is the cyclomatic number of G. In this paper we find an exact expression for Szeged and Balaban indices of TUHC6[2p, q], the zigzag polyhex nanotubes, using a theorem of Dobrynin and Gutman on connected bipartite graphs (see Ref [11]).
