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...

Expressions for the Zagreb indices and coindices of the total graph, semi-total point graph and of semi-total line graph of subdivision graphs in terms of the parameters of the parent graph are obtained, thus generalizing earlier existing results.

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...

A connected graph G is said to be neighbourly irregular graph if no two adjacent vertices of G have same degree. In this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

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...