Let G = (V,E) be a graph. By directional labeling (or d-labeling) of an edge x = uv of G by an ordered n-tuple (a1, a2, · · · , an), we mean a labeling of the edge x such that we consider the label on uv as (a1, a2, · · · , an) in the direction from u to v, and the label on x as (an, an−1, · · · , a1) in the direction from v to u. In this survey, we study graphs, called (n, d)-sigraphs, in whic...

We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g−1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less ...

A two-valued function f defined on the vertices of a graph G (V, E), I : V -+ {-I, I}, is a signed dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v]) 2: 1, where N(v] consists of v and every vertex adjacent to v. The of a signed dominating function is ICV) = L f( v), over all vertices v E V. The signed domination...

A signed tree-coloring of a signed graph (G, σ) is a vertex coloring c so that G(i,±) is a forest for every i ∈ c(u) and u ∈ V(G), where G(i,±) is the subgraph of (G, σ) whose vertex set is the set of vertices colored by i or −i and edge set is the set of positive edges with two end-vertices colored both by i or both by −i, along with the set of negative edges with one end-vertex colored by i a...

We study a class of hyperplane arrangements associated to complementary signed graphs in which the positive part and the negative part are complementary to each other in Kn, the complete graph on n vertices. These arrangements form a subclass of the Dn arrangement but do not contain the An−1 arrangement. The main result says that the arrangement A(G) of a complementary signed graph G is superso...

We consider a graph-theoretical formalization of the process of gene assembly in ciliates introduced in Ehrenfeucht et al (2003), where a gene is modeled as a signed graph. The gene assembly, based on three types of operations only, is then modeled as a graph reduction process (to the empty graph). Motivated by the robustness of the gene assembly process, the notions of parallel reduction and p...

Motivated by the notion of single valued neutrosophic graphs defined by Broumi, Talea, Bakali and Smarandache[2] and notion of intuitionistic fuzzy signed graphs defined by Mishra and Pal[8], we introduce the concept of single valued neutrosophic signed graphs and examine the properties of this new concept and examples.

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties edge-coloured graphs. A homomorphism from $G$ $H$ is a vertex-mapping that preserves adjacencies and edge-colours. consider property having fixed $H$. The question we are interested in is: given an $G$, can perform $k$ operations so resulting admits $H$? vertex-deletion, edge-deletion ...

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative 2-clique with any positive (n, 3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. Next, we prove that a s...

This survey paper provides an introduction to signed graphs, focusing on coloring. We shall introduce the concept of signed graphs, a proper coloring, and basic properties, such as a balanced graph and switchings. We will examine the chromatic number for six special signed graphs, upper bound the chromatic number, and discuss practical applications of signed graphs.