Randi\'c index, diameter and the average distance

The Randi\'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi\'c index $R(G)$ with relations to the diameter $D(G)$ and the average distance $\mu(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $\delta(G)$, if $\delta(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $\delta(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq \mu(G)$. Furthermore, for any arbitrary real number $\varepsilon \ (0<\varepsilon<1)$, if$\delta(G)\geq \varepsilon n$, then$\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$and$R(G)\geq \mu(G)$hold for sufficiently large$n$.Comment: 7 page

The asymptotic value of Randic index for trees

Let $\mathcal{T}_n$ denote the set of all unrooted and unlabeled trees with $n$ vertices, and $(i,j)$ a double-star. By assuming that every tree of $\mathcal{T}_n$ is equally likely, we show that the limiting distribution of the number of occurrences of the double-star $(i,j)$ in $\mathcal{T}_n$ is normal. Based on this result, we obtain the asymptotic value of Randi\'c index for trees. Fajtlowicz conjectured that for any connected graph the Randi\'c index is at least the average distance. Using this asymptotic value, we show that this conjecture is true not only for almost all connected graphs but also for almost all trees.Comment: 12 page

Spectra of eccentricity matrices of graphs

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the least eigenvalue of eccentricity matrices of trees, presented in the article [Jianfeng Wang, Mei Lu, Francesco Belardo, Milan Randic. The anti-adjacency matrix of a graph: Eccentricity matrix. Discrete Applied Mathematics, 251: 299-309, 2018.], is solved affirmatively. In addition to this, the spectra and the inertia of eccentricity matrices of various classes of graphs are investigated.Comment: Comments are welcome

Efficient computation of trees with minimal atom-bond connectivity index

The {\em atom-bond connectivity (ABC) index} is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph $G$, the ABC index is defined as $\sum_{uv\in E(G)}\sqrt{\frac{(d(u) +d(v)-2)}{d(u)d(v)}}$, where $d(u)$ is the degree of vertex $u$ in $G$ and $E(G)$ is the set of edges of $G$. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal $ABC$ index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of the graphs with minimal $ABC$ index. The obtained results disprove some existing conjectures end suggest new ones to be set

Computational and analytical studies of the Randi\'c index in Erd\"os-R\'{e}nyi models

In this work we perform computational and analytical studies of the Randi\'c index $R(G)$ in Erd\"os-R\'{e}nyi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $\xi\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $\xi < 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < \xi < 10$ (where $0 10$ ($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle \overline{R}(G) \right\rangle$, we analytically (i) obtain new relations connecting $R(G)$ with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on $R(G)$ for graphs in Erd\"os-R\'{e}nyi models.Comment: 28 pages, 10 figure

On structural properties of trees with minimal atom-bond connectivity index

The {\em 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 reveal that trees with minimal ABC index do not contain so-called {\em $B_k$-branches}, with $k \geq 5$, and that they do not have more than four $B_4$-branches

A survey on the skew energy of oriented graphs

Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. The energy of $G$ is defined as the sum of absolute values of all eigenvalues of $A(G)$, which was introduced by Gutman in 1970s. Since graph energy has important chemical applications, it causes great concern and has many generalizations. The skew energy and skew energy-like are the generalizations in oriented graphs. Let $G^\sigma$ be an oriented graph of $G$ with skew adjacency matrix $S(G^\sigma)$. The skew energy of $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the norms of all eigenvalues of $S(G^\sigma)$, which was introduced by Adiga, Balakrishnan and So in 2010. In this paper, we summarize main results on the skew energy of oriented graphs. Some open problems are proposed for further study. Besides, results on the skew energy-like: the skew Laplacian energy and skew Randi\'{c} energy are also surveyed at the end.Comment: This will appear as a chapter in Mathematical Chemistry Monograph No.17: Energies of Graphs -- Theory and Applications, edited by I. Gutman and X. L

Maximizing spectral radius and number of spanning trees in bipartite graphs

The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance distance between the end-vertices of another edge

Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphs

Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph $G(m,n,p)=K_{m,n}-pK_2$ $(p\leq \min\{m,n\})$, which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in $G(n,n,p)$. As a corollary, we obtain the Kirchhoff index of $G(m,n,p)$ which extends a previous result by Shi and Chen

The Sierpi\'nski product of graphs

In this paper we introduce a product-like operation that generalizes the construction of generalized Sierpi\'nski graphs. Let $G,H$ be graphs and let $f: V(G) \to V(H)$ be a function. Then the Sierpi\'nski product of $G$ and $H$ with respect to $f$ is defined as a pair $(K,\varphi)$, where $K$ is a graph on the vertex set $V(G) \times V(H)$ with two types of edges: -- $\{(g,h),(g,h')\}$ is an edge in $K$ for every $g\in V(G)$ and every $\{h,h'\}\in E(H)$, -- $\{(g,f(g'),(g',f(g))\}$ is an edge in $K$ for every edge $\{g,g'\} \in E(G)$; and $\varphi: V(G) \to V(K)$ is a function that maps every vertex $g \in V(G)$ to the vertex $(g,f(g)) \in V(K)$. Graph $K$ will be denoted by $G\otimes_f H$. Function $\varphi$ is needed to define the product of more than two factors. By applying this operation $n$ times to the same graph we obtain the $n$-th generalized Sierpi\'nski graph. Some basic properties of the Sierpi\'nski product are presented. In particular, we show that $G \otimes_f H$ is connected if and only if both $G$ and $H$ are connected and we present some necessary and sufficient conditions that $G,H$ must fulfill in order for $G \otimes_f H$ to be planar. As for symmetry properties, we show which automorphisms of $G$ and $H$ extend to automorphisms of $G \otimes_f H$. In many cases we can also describe the whole automorphism group of $G\otimes_f H$