Graphs and matrices with maximal energy
نویسنده
چکیده
Given a complex m n matrix A; we index its singular values as 1 (A) 2 (A) ::: and call the value E (A) = 1 (A)+ 2 (A)+ ::: the energy of A; thereby extending the concept of graph energy, introduced by Gutman. Koolen and Moulton proved that E (G) (n=2) (1 + p n) for any graph G of order n and exhibited an in nite family of graphs with E (G) = (v (G) =2) 1 + p v (G) . We prove that for all su¢ ciently large n; there exists a graph G = G (n) with E (G) n3=2=2 n11=10. This implies a conjecture of Koolen and Moulton. We also characterize all square nonnengative matrices and all graphs with energy close to the maximal one: In particular, such graphs are quasi-random.
منابع مشابه
Some Results on the Maximal 2-Rainbow Domination Number in Graphs
A 2-rainbow dominating function ( ) of a graph is a function from the vertex set to the set of all subsets of the set such that for any vertex with the condition is fulfilled, where is the open neighborhood of . A maximal 2-rainbow dominating function on a graph is a 2-rainbow dominating function such that the set is not a dominating set of . The weight of a maximal is the value . ...
متن کاملLine graphs associated to the maximal graph
Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...
متن کاملOn the Fischer-Clifford matrices of a maximal subgroup of the Lyons group Ly
The non-split extension group $overline{G} = 5^3{^.}L(3,5)$ is a subgroup of order 46500000 and of index 1113229656 in Ly. The group $overline{G}$ in turn has L(3,5) and $5^2{:}2.A_5$ as inertia factors. The group $5^2{:}2.A_5$ is of order 3 000 and is of index 124 in L(3,5). The aim of this paper is to compute the Fischer-Clifford matrices of $overline{G}$, which together with associated parti...
متن کاملOn the saturation number of graphs
Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation numbe...
متن کاملF-Permutations induce Some Graphs and Matrices
In this paper, by using the notion of fuzzy subsets, the concept of F-permutation is introduced. Then by applying this notion the concepts of presentation of an F-polygroup, graph of an F-permutation and F-permutation matrices are investigated.
متن کامل