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 survey the recent work on phase transition and distances in various random graph models with general degree sequences. We focus on inhomogeneous random graphs, the configuration model and affine preferential attachment models, and pay special attention to the setting where these random graphs have a power-law degree sequence. This means that the proportion of vertices with degree k in large ...

Let $G*H$ be the product $*$ of $G$ and $H$. In this paper we determine the rth power of the graph $G*H$ in terms of $G^r, H^r$ and $G^r*H^r$, when $*$ is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation $(G*H)^r=G^r*H^r$. We also compute the Wiener index and Wiener polarity index of the skew product.

This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent with degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal to use the seeds recover remaining latent vertex correspondence between Departing existing approaches focus on high-d...

A graph G that has a perfect matching is n-extendable if every matching of size n lies in a perfect matching of G. We show that when the connectivity of a line graph, power graph, or total graph is sufficiently large then it is n-extendable. Specifically: if G has even size and is (2n + 1)edge-connected or (n + 2)-connected, then its line graph is n-extendable; if G has even order and is (n + 1...

An adjacency labeling scheme labels the n nodes of a graph with bit strings in a way that allows, given the labels of two nodes, to determine adjacency based only on those bit strings. Though many graph families have been meticulously studied for this problem, a non-trivial labeling scheme for the important family of power-law graphs has yet to be obtained. This family is particularly useful fo...

We consider the classical push broadcast process on a large class of sparse random multigraphs that includes random power law graphs and multigraphs. Our analysis shows that for every ε > 0, whp O(log n) rounds are sufficient to inform all but an ε-fraction of the vertices. It is not hard to see that, e.g. for random power law graphs, the push process needs whp n rounds to inform all vertices. ...

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 2002, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every...

An adjacency labeling scheme labels the n nodes of a graph with bit strings in a way that allows, given the labels of two nodes, to determine adjacency based only on those bit strings. Though many graph families have been meticulously studied for this problem, a non-trivial labeling scheme for the important family of power-law graphs has yet to be obtained. This family is particularly useful fo...

The aim of the paper is to develop a logic of relations on semiconcept graphs corresponding to the Contextual Logic of Relations on power context families. Semiconcept graphs allow the representation of negations. The operations from Peircean Algebraic Logic (i.e., the operations of relation algebras of power context families) are used to generate compound semiconcepts (or relations, resp.). Fo...