Related papers: Short note on Randi\'{c} energy
The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of…
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…
We review bounds for the general Randi\'c index, $R_{\alpha} = \sum_{ij \in E} (d_i d_j)^\alpha$, and use the power mean inequality to prove, for example, that $R_\alpha \ge m\lambda^{2\alpha}$ for $\alpha < 0$, where $\lambda$ is the…
The graph having the minimum reduced reciprocal Randi\'c index is characterized among the class of all unicyclic graphs with fixed number of vertices.
We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…
A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $ is a graph where the gain function $ \varphi $ assigns a unit complex number to each orientation of an edge of $ G $ and its inverse is assigned to the opposite…
Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching…
For $l > 1$, the $l$-edge-connectivity $\kappa'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if…
O and Shi proved that the Randi\'c index of any graph $G$ with minimum degree at least $\delta$ and maximum degree at most $\Delta$ is at least $\frac{\sqrt{\delta\Delta}}{\delta+\Delta}|G|$, with equality if and only if the graph is…
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…
We establish a lower bound for the energy of a complex unit gain graph in terms of the matching number of its underlying graph, and characterize all the complex unit gain graphs whose energy reaches this bound.
We investigate some topological and spectral properties of Erd\H{o}s-R\'{e}nyi (ER) random digraphs $D(n,p)$. In terms of topological properties, our primary focus lies in analyzing the number of non-isolated vertices $V_x(D)$ as well as…
In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
Summary statistics play an important role in network data analysis. They can provide us with meaningful insight into the structure of a network. The Randi\'{c} index is one of the most popular network statistics that has been widely used…
In this paper, we mainly study the trace norm of the adjacency matrix of a graph, also known as the energy of graph. We give the maximum trace norms for the graph and its complement. In fact, the above problem is stated and solved in a more…
We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately…
The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as $v_1, v_2, \ldots, v_n$, so that for each $2 \le i \le n$, vertex $v_i$ is either adjacent…
A graph is trivial if it contains one vertex and no edges. The essential connectivity $\kappa^{\prime}$ of $G$ is defined to be the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two non-trivial…
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance…