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...

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and –1 such that the closed neighbourhood of every vertex contains more +1’s than –1’s. This concept is closely related to combinatorial discrepancy theory as shown by Füredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223–239]. The signed domination number of G is the minimum of the sum...

Let D = (V,A) be a finite simple directed graph (shortly digraph) in which dD(v) ≥ 1 for all v ∈ V . A function f : V −→ {−1, 1} is called a signed total dominating function if ∑ u∈N−(v) f(u) ≥ 1 for each vertex v ∈ V . A set {f1, f2, . . . , fd} of signed total dominating functions on D with the property that ∑d i=1 fi(v) ≤ 1 for each v ∈ V (D), is called a signed total dominating family (of f...

The isomorphism classes of several types of graph coverings of a graph have been enumerated by many authors [M. Hofmeister, Graph covering projections arising from finite vector space over finite fields, Discrete Math. 143 (1995) 87–97; S. Hong, J.H. Kwak, J. Lee, Regular graph coverings whose covering transformation groups have the isomorphism extention property, Discrete Math. 148 (1996) 85–1...

We introduce joins of signed graphs and explore the chromatic number all-positive all-negative joins. prove an analogue to theorem that join two equals sum their numbers. Given graphs, is usually less than numbers, by amount depends on new concept deficiency a signed-graph coloration.

The open neighborhood NG(e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e and its closed neighborhood is NG[e] = NG(e) ∪ {e}. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If ∑x∈NG[e] f(x) ≥ 1 for at least a half of the edges e ∈ E(G), then f is called a signed edge majority dominating function of G. The minimum of the val...

A (m,n)-colored mixed graph is a graph having arcs of m different colors and edges of n different colors. A graph homomorphism of a (m,n)-colored mixed graph G to a (m,n)colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (m,n)-colored mixed chromatic number χ(m,n)(G) of a (m,n)-colored mixed graph G, in...

Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to (including itself) is positive. Then weights over at least $-\frac{n^2}{25}$. Also we provide an example weighted described properties $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$.
 The previous best known bounds were $-\frac{n^2}{16}$ $-(1+o(1))\frac{n^2}{54}$...

The paper primarily tends to handle bipartite output regulation problem for singular heterogeneous multi-agent systems with signed graph. This disturbance and reference signals are produced by exosystem, respectively. By virtue of theory, two new distributed feedback controllers state considered realize the regulation, For each case, a dynamical compensator is proposed, stability analysis close...

Abstract As a fundamental data structure, graphs are ubiquitous in various applications. Among all types of graphs, signed bipartite contain complex structures with positive and negative links as well settings, on which conventional graph analysis algorithms no longer applicable. Previous works mainly focus unipartite or unsigned separately. Several models proposed for applications the by utili...