Related papers: Energy of a graph and Randic index

The energy of a graph is the sum of the absolute values of its adjacency eigenvalues. For integral circulant graphs $\ICG(n,\mathcal{D})$ of order $n=p^2q^3$, where $p$ and $q$ are distinct odd primes, we prove that the adjacency…

In this paper, we give several constructions for the pairs of graphs to be equienergetic and Randic' equienergetic graphs. Also, some new families of integral and Randic' integral graphs are obtained. As an application, a sequence of graphs…

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we present a new method to compare the energies of two $k$-subdivision bipartite graphs on some cut edges. As the…

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,..., v_n\}$. The Randi\'{c} matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_id_j)^{\frac{-1}{2}}$ if $v_i$ and $v_j$ are adjacent…

For a simple graph $G$ with adjacency matrix $A(G)$, let $\pi(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $\mu_1,\ldots,\mu_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n…

In this paper, first we compute the energy of a special partitioned matrix under some cases. As a consequence, we obtain the reciprocal distance energy of the complete multipartite graph and also we give various other energies of complete…

Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by, $\mathcal{E}(G)$ is the sum of all absolute values of the eigenvalues of the adjacency matrix $A(G)$. It is the first eigenvalue-based topological…

A graph G is said to be orderenergetic, if its energy equal to its order and it is said to be hypoenergetic if its energy less than its order. Two non-isomorphic graphs of same order are said to be equienergetic if their energies are equal.…

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $\cal D$ of…

The power graph of an arbitrary group $G$ is a simple graph with all elements of $G$ as its vertices and two vertices are adjacent if one is a positive power of another. In this paper, we generalize this concept to a graph whose vertices…

The \emph{Antimagic Graph Conjecture} asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, ..., |E|$ is used exactly once and the sums of the labels on all edges incident with a…

The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $n$ be an even number and $\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$ with maximum…

Motivated by the linear time algorithm that locates the eigenvalues of a cograph G [10], we investigate the multiplicity of eigenvalue for \lambda \neq -1,0. For cographs with balanced cotrees we determine explicitly the highest value for…

We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through…

The energy of a vertex $v_i$ in a graph $G$ is defined as $\mathcal{E}_G(v_i) = |A|_{ii}$, where $A$ is the adjacency matrix of $G$, $A^*$ denotes the conjugate transpose of $A$, and $|A| = (AA^*)^{1/2}$. The total energy of the graph,…

Graph energy and Domination in graphs are most studied areas of graph theory. In this paper we made an attempt to connect these two areas of graph theory by introducing c-dominating energy of a graph $G$. First, we show the chemical…

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any…

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.

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$…

Let G = G(V,E) be a graph with | V | vertices and | E | edges and total graph, T(G) is obtained from G. In This paper we have study the Harmonic index of total graph for standards graphs, bipartite graph of particular type, regular graph,…