Narrow Results

The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes.

Let $G$ be a graph with edge set $E(G)$. For an edge $e=uv$ in $G$, we define $p{d}_G(e)=d_G(u)d_G(v)$, where $d_G(u)$ and $d_G(v)$ are degrees of vertices $u$ and $v$ in $G$, respectively. For $G$, the graph invariants $E{M}^{pd}$, $E{M}_1^{pd}$ and $E{M}_2^{pd}$ are defined as $E{M}^{pd}(G)={\sum }_{e\in E(G)}p{d}_G{(e)}^2$, $E{M}_1^{pd}(G)={\sum }_{e\sim f}[p{d}_G(e)+p{d}_G(f)]$ and $E{M}_2^{pd}(G)={\sum }_{e\sim f}p{d}_G(e)p{d}_G(f)$, where $e\sim f$ means that the edges $e$ and $f$ are incident. In this paper, some relationship between these graph invariants and some classical topological indices were presented. Moreover, some bounds for $E{M}^{pd}$, $E{M}_1^{pd}$ and $E{M}_2^{pd}$ are obtained and trees with the first through the third smallest $E{M}^{pd}$ and $E{M}_1^{pd}$, as well as the trees with the first through the forth smallest $E{M}_2^{pd}$ are also characterized.

For a graph, the first (multiplicative) Zagreb index is equal to the sum (product) of squares of the vertex degrees, and the second (multiplicative) Zagreb index is equal to the sum (product) of products of the degrees of a pair of adjacent vertices. In this work, by a unified approach, we determine the extremal values of these Zagreb indices in terms of the (edge) connectivity and characterize the corresponding extremal graphs among all connected bipartite graphs of order $n$. Our results show that the extremal graphs of given (edge) connectivity regarding the Zagreb indices and multiplicative Zagreb indices do not completely coincide with other topological indices.

For a $k$-uniform hypergraph $H$, we introduce degree-based indices such as the general sum-connectivity index ${\chi }_a(H)={\sum }_{v_1{v}_2\cdots v_k\in E(H)}{[d(v_1)+d(v_2)+\cdots +d(v_k)]}^a$ and the general Randić index $R_a(H)={\sum }_{v_1{v}_2\cdots v_k\in E(H)}{[d(v_1)d(v_2)\cdots d(v_k)]}^a$, where $a\in ℝ$, $E(H)$ is the set of hyperedges of $H$ and $d(v_i)$ is the degree of a vertex $v_i$ in $H$; $k\ge 2$ and $i=1,2,\dots ,k$. Other indices such as the first and second Zagreb index, first and second hyper-Zagreb index, classical sum-connectivity index, classical Randić index and harmonic index of a hypergraph $H$ are special cases of the general indices. For $a>0$, we obtain upper bounds on ${\chi }_a(H)$ and $R_a(H)$ for a uniform hypergraph $H$ with given order, order and number of isolated vertices, order and maximum degree, order and diameter at least $2$, and lower bounds for uniform hypergraphs with given order and no isolated vertices, order and minimum degree, and order and maximum possible degree. We also present extremal graphs for all the bounds. Bounds on Zagreb indices follow from our results on general indices.

We determine the exact value of the degree distance of the cartesian product and the wreath product of two graphs. Also we apply our results to compute the exact degree distance of hypercube, torus, grid, Hamming graph and fence graph.

The exact value of the Gutman indices of the Cartesian product and the wreath product of graphs are obtained. Using the results obtained here, the exact Gutman indices of the hypercube, torus, grid, Hamming graph and the fence graph are given.

In this paper we construct a Cayley graph for multiplicative group of upper unitriangular $2\times 2$ matrices over $z$ mod $n$. Also we find some topological indices, diameter, girth, spectra and energy of adjacency, Laplacian, normalized Laplacian and signless Laplacian matrix of the same graph.

Let $\prod (G)$ be multiplicative Zagreb index of a graph $G$. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of $\prod (G)$ for all cactus graphs and characterize the corresponding extremal graphs.

The topological indices are useful tools to the theoretical chemists that are provided by the graph theory. They correlate certain physicochemical properties such as boiling point, strain energy, stability, etc. of chemical compounds. For a graph $G$, the double graph $D[G]$ is a graph obtained by taking two copies of graph $G$ and joining each vertex in one copy with the neighbors of corresponding vertex in another copy and strong double graph SD$[G]$ of the graph $G$ is the graph obtained by taking two copies of the graph $G$ and joining each vertex $v$ in one copy with the closed neighborhood of the corresponding vertex in another copy. In this paper, we compute the general sum-connectivity index, general Randi$ć$ index, geometric–arithmetic index, general first Zagreb index, first and second multiplicative Zagreb indices for double graphs and strong double graphs and derive the exact expressions for these degree-base topological indices for double graphs and strong double graphs in terms of corresponding index of original graph $G$.

The first Zagreb index of a graph $G$ is the sum of squares of the degrees of the vertices of $G$. In this paper, we introduce generalized four new sums of graphs and study the first Zagreb index and coindex of the resulting graphs. In addition, we give the short proof for the earlier results of Deng, Sarala, Ayyaswamy and Balachandran [Appl. Math. Comput. 275 (2016) 422–431] on the first Zagreb index of four operations on graphs by different approach.

The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph. Entropy functions have been used successfully to capture different aspects of graph complexity. The generalized graph entropies result from applying information measures to a graph using various schemes for defining probability distributions over the elements of the graph. In this paper, we investigate the complexity of a class of composite graphs based on subdivision graphs and corona product evaluating the generalized graph entropies, and we present explicit formulae for the complexity of subdivision-corona type product graphs.