For a d-tuple of commuting operators S := (S1,..., Sd) ? B[X]d, m N and p (0,?), we define Q(p) (S;u) 0?k?m (-1)k (m k) (???Nd0 |?| = k k!/? ||S?u||p). As natural extension the concepts (m,p)-expansive (m,p)-contractive for tuple operators, introduce study (m,?)-expansive (m,?)-contractive acting on Banach space. We say that is (resp. (m,?)- contractive d-tuple) if Q(p)m 0 u X ?) . These extend...

Let K n denote the Cartesian product Kn Kn Kn, where Kn is the complete graph on n vertices. We show that the domination number of K n

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. The power domination problem is a variant of the classical domination problem in graphs and is defined as follows. Given an undirected graph G = (V, E), the problem is to find a minimum vertex set SP ⊆ V , called the power dominating set of G, such that all vertices in G are ob...

A k-dominating set of a graph G is a set S of vertices of G such that every vertex outside of S has k neighbors in S. The k-domination number of G, written γk(G), is the size of the smallest k-dominating set in G. In this paper, we derive sharp upper and lower bounds on γk(G) + γk(G) and γk(G)γk(G), where G is the complement of G. We use the results for k = 2 to prove a conjecture of Alon, Balo...

a {em roman dominating function} on a graph $g$ is a function$f:v(g)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}a {em restrained roman dominating}function} $f$ is a {color{blue} roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} the wei...

(1) an n+1-tuple (ρ, χ1, ...., χn) of nontrivial C×-valued multiplicative characters of k×, each extended to k by the requirement that it vanish at 0 ∈ k. (2) an n+1-tuple (g, f1, ...., fn) of nonzero one-variable k-polynomials, which are adapted to the character list above in the following sense. Whenever α ∈ k is a zero of g (respectively of some fi), then ρα (respectively χ ordα(fi) i ) is n...

This thesis deals with the following three independent problems. Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgra...