Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f...

We investigate a domination-like problem from the exact exponential algorithms viewpoint. The classical Dominating Set problem ranges among one of the most famous and studied NP -complete covering problems [6]. In particular, the trivial enumeration algorithm of runtime O∗(2n) 4 has been improved to O∗(1.4864n) in polynomial space, and O∗(1.4689n) with exponential space [9]. Many variants of th...

Let G^s be a signed graph, where G = (V;E) is the underlying simple graph and s : E(G) to {+, -} is the sign function on E(G). In this paper, we obtain k-th signed spectral moment and k-th signed Laplacian spectral moment of Gs together with coefficients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.

The total graph is built by joining the to its line means of incidences. We introduce a similar construction for signed graphs. Under two defnitions graph, we defne corresponding and show that it stable under switching. consider balance, frustration index number, largesteigenvalue. In regular case compute spectrum adjacency matrix spectra certain compositions, determine some with exactly main e...

Definition of dominating function on a fractional graph G has been introduced. Fractional parameters such as domination number and upper defined. Domination with fuzzy Intuitionistic environment, have found by formulating Linear Programming Problem.

The Moore-Penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs. For such a matrix, the sign of every entry of the Moore-Penrose inverse is shown to be determined uniquely by the signs of the matrix entries; i.e., the matrix has a signed generalized inverse. Necessary and sufficient conditions on an acyclic bipartite gra...

This paper is motivated by the concept of nonnegative signed domination that was introduced by Huang, Li, and Feng in 2013 [15]. We study the non-negative signed domination problem from the theoretical point of view. For networks modeled by strongly chordal graphs and distance-hereditary graphs, we show that the non-negative signed domination problem can be solved in polynomial time. For networ...

Let G = ( V, E ) be a simple graph with vertex setxs V and edge set . A mixed Roman dominating function of is f : ∪ → {0, 1, 2} satisfying the condition that every element x ∈ for which f(x) 0 adjacent or incident to at least one y f(y) 2. The weight ω( ∑ domination number γ R minimum We first show problem computing * NP-complete bipartite graphs then we present upper lower bounds on number, so...