More on energy and Randic energy of specific graphs

Let $G$ be a simple graph of order $n$. The energy $E(G)$ of the graph $G$ is the sum of the absolute values of the eigenvalues of $G$. 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 and $0$ for another cases. The Randi\'{c} energy $RE$ of $G$ is the sum of absolute values of the eigenvalues of $R(G)$. In this paper we compute the energy and Randi\'{c} energy for certain graphs. Also we propose a conjecture on Randi\'c energy.Comment: 14 page

Introduction to end super dominating sets in graphs

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. In this paper, we define a new domination number, and call it end super domination number. We give some applications of this definition and obtain the exact value of that on specific graphs. We count the number of end super dominating sets of these graphs too. Also, we present some sharp bounds on the end super domination number, where graph is modified by vertex (edge) removal and contraction. Finally, we generalize our definition and present some results on that.Comment: 22 pages, 10 figure

Secure domination number of kk-subdivision of graphs

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called a secure dominating set of $G$, if for every $u\in V-D$, there exists a vertex $v\in D$ such that $uv \in E$ and $D-\{v\}\cup\{u\}$ is a dominating set of $G$. The cardinality of a smallest secure dominating set of $G$, denoted by $\gamma_s(G)$, is the secure domination number of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the secure domination number of $k$-subdivision of $G$.Comment: 10 Pages, 8 Figure

On the Sombor characteristic polynomial and Sombor energy of a graph

Let G be a simple graph with vertex set V(G)={v1,v2,…,vn}. The Sombor matrix of G, denoted by ASO(G), is defined as the n×n matrix whose (i, j)-entry is d2i+d2j−−−−−−√ if vi and vj are adjacent and 0 for another cases. Let the eigenvalues of the Sombor matrix ASO(G) be ρ1≥ρ2≥⋯≥ρn which are the roots of the Sombor characteristic polynomial ∏ni=1(ρ−ρi). The Sombor energy ESO of G is the sum of absolute values of the eigenvalues of ASO(G). In this paper, we compute the Sombor characteristic polynomial and the Sombor energy for some graph classes, define Sombor energy unique and propose a conjecture on Sombor energy.publishedVersio