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.

Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the pro...

Let G = (V, E) be a graph. A function f : V → {−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, γ t (G), is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen...

Abstract. Let G = (V,E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ0 st(G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on...

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

A nonnegative signed dominating function (NNSDF) of a graph G is a function f from the vertex set V (G) to the set {−1, 1} such that ∑ u∈N [v] f(u) ≥ 0 for every vertex v ∈ V (G). The nonnegative signed domination number of G, denoted by γ s (G), is the minimum weight of a nonnegative signed dominating function on G. In this paper, we establish some sharp lower bounds on the nonnegative signed ...

Let G = (V, E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. Let G denote the complement of G. In this pa...

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman k-dominating function (STRkDF) on D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−(v) f(x) ≥ k for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight o...

A signed Roman dominating function on the digraphD is a function f : V (D) −→ {−1, 1, 2} such that ∑ u∈N−[v] f(u) ≥ 1 for every v ∈ V (D), where N−[v] consists of v and all inner neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an inner neighbor v for which f(v) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman dominating functions on D with the property that ∑d i=1 fi(...