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.
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.
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.
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.
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.
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.
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.
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 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).
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.
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]).
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.
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.
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]).
