Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...

We introduce a new topological descriptor of a network called the density decomposition which is a partition of the nodes of a network into regions of uniform density. The decomposition we define is unique in the sense that a given network has exactly one density decomposition. The number of nodes in each partition defines a density distribution which we find is measurably similar to the degree...

A proposal is presented for simulating an improvement on quenched QCD with dynamical fermions which interact with the gluon configuration only via the topological index of the latter. Strengths and shortcomings of the method are discussed and it is argued that the approximation – though being crude – shares some qualitative aspects of full QCD which relate to the issue of chiral symmetry breaking.

Let C r ((0; 1]) denote the metric space of C r self-maps of 0; 1] and C s;r ((0; 1]) denote the metric space of C s families of maps in C r ((0; 1]) with the parameter space 0; 1]. Let Hs;r be the unimodal families with non-constant topological entropy in C s;r ((0; 1]). We show that for s 0, r 2, there is an open and dense subset Gs;r of Hs;r such that each family in Gs;r has a map with a hom...

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G)  V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investiga...

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results r...

We prove an index theorem for boundary value problems in Boutet de Monvel’s calculus on a compact manifold X with boundary. The basic tool is the tangent semigroupoid T −X generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C∗ r (T −X) of C∗-algebras over [0, 1]. Its fiber in ~ = 0, C∗ r (T−X), can be identified with the symbol algebr...

The connectivity index χ1(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T (n, r) be the set of trees on n vertices with diameter r . In this paper, we determine all trees in T (n, r) with the largest and the second largest connectivity index. Also, the trees in T (n, r) with the largest and the second largest connecti...

The atom-bond connectivity (ABC) index is a recently introduced topological index, defined as ABC(G) = ∑ uv∈E(G) √ du+dv−2 dudv , where du (or dv) is the degree the vertex u (or v). The ABC index of benzenoid systems and phenylenes are computed, a simple relation is established between the atom-bond connectivity index of a phenylene and the corresponding hexagonal squeeze in this paper. Mathema...

The entropy of a graph was first introduced by Rashevsky [1] and Trucco [2] to interpret as the structural information content of the graph and serve as a complexity measure. In this paper, we first state a number of definitions of graph entropy measures and generalized graph entropies. Then we survey the known results about them from the following three respects: inequalities and extremal prop...