In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also stu...

In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets of nontrivial elements in G. We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply conn...

A graph $G$ is called $P_4$-free, if $G$ does not contain an induced subgraph $P_4$. The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,x)$ is called a domination root of $G$. In this paper we state and prove formula for the domination polynomial of no...

The corona G◦H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining each i-th vertex of G to every vertex in the i-th copy of H. The neighborhood coronaG⋆H of two graphs G and H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. In this p...

Let Th be the complete binary tree of height h. Let M be the infinite grid graph with vertexset Z, where two vertices (x1,y1) and (x2,y2) of M are adjacent if and only if|x1-x2| + |y1-y2| = 1. Suppose that T is a tree which is a subdivision of Th and is also isomorphicto a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansionration(T)<lb...

Abstract We give the equations of n -th symmetric product $$X^n/S_n$$ X n / S a flat affine scheme $$X=\mathrm {Spec}\,A$$ = Spec A</...

The focus of the present work was on the association between commuting (business and private), life satisfaction, stress, and (over-) use of the Internet. Considering that digital devices are omnipresent in buses and trains, no study has yet investigated if commuting contributes to the development of Internet addiction. Overall, N = 5039 participants (N = 3477 females, age M = 26.79, SD = 10.68...

Let G = (V (G), E(G)) be a simple graph, and let A be its adjacency matrix. The characteristic polynomial det(xI −A) of A is called the characteristic polynomial of G, and is denoted by φ(G, x). The eigenvalues of A (i.e. the zeros φ(G, x)) are called the eigenvalues of G. The index of a graph G is the largest eigenvalue of G, denoted by ρ(G). It has been studied extensively in the literature [...

let $g$ be a finite group‎. ‎a subset $x$ of $g$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $x$ do not commute‎. ‎in this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.