On general sum-connectivity index

We report some properties of the reverse degree distance of a connected (molecular) graph, and, in particular, its relationship with the first Zagreb index and Wiener index. We also show that the reverse degree distance satisfies the basic requirement for a branching index.     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

Minimal energy of unicyclic graphs of a given diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. For a given positive integer d with , we characterize the graphs with minimal energy in the class of unicyclic graphs with n vertices and a given diameter d.      

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

Some results on energy of unicyclic graphs with n vertices

A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Hou (J Math Chem 29:163–168, 2001) first considered the minimal energy for general unicyclic graphs. In this paper, we determine the unicyclic graphs with the minimal energy in U_n^l{\mathcal {U}_n^l} and the unicyclic graphs with the first forth smallest energy in U_n (n 3 13){\mathcal {U}_n\,(n\geq 13)} vertices.     

The extremal structures of k-uniform unicyclic hypergraphs on Wiener index

The Wiener index of a connected k-uniform hypergraph is defined as the summation of distances between all pairs of vertices. We determine the unique k-uniform unicyclic hypergraphs with maximum and second maximum, minimum and second minimum Wiener indices, respectively.     

Harary index of Eulerian graphs

Journal of Mathematical Chemistry - The Harary index of a connected (molecular) graph is defined as the sum of reciprocals of distances between all its pairs of vertices. In this paper, we...     

On the Kirchhoff index of regular graphs

Using probabilistic tools, we give a compact formula for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and Kemeny's constant, as well as general upper and lower bounds in terms of d and N. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

CMMSE-on the first general Zagreb index

The aim of this paper is to obtain new inequalities involving the first general Zagreb index, and characterize graphs which are extremal with respect to them. We also obtain inequalities involving the forgotten and second general Zagreb indices.     

Three methods for calculation of the hyper-Wiener index of molecular graphs

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.     

广义α~N指数及其应用 IV: 广义α~N指数在碳氢化合物中的应用

本文计算了七对顺、反异构体的广义α~N指数, 结果表明, 广义α~N指数不仅能很好地区分碳氢化合物的顺、反异构体, 而且与其物理性质(沸点、密度、折光率、分子折射度等)有良好的单变量线性相关关系, 解决了拓扑指数不能区分顺、反异构体的困难。本文还计算了庚烷、辛烷系列的广义α~N指数, 并比较了多种拓扑指数与辛烷值的相关分析结果。     

On the ABC index of connected graphs with given degree sequences

The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence.     

Important mathematical structures of the topological index Z for tree graphs

Mathematical importance of the topological index, ZG, or the so-called Hosoya index is stressed by presenting and giving supporting evidence for the proposed conjecture. That is, for a given pair of positive integers (n1or=3), with Z(G1) = n1 and Z(G2) = n2.