Common Neighborhood Graph

The Common Neighborhood Graph and Its Energy

Let $G$ be a simple graph with vertex set ${v_1,v_2,ldots,v_n}$. The common neighborhood graph (congraph) of $G$, denoted by $con(G)$, is the graph with vertex set ${v_1,v_2,ldots,v_n}$, in which two vertices are adjacent if and only they have at least one common neighbor in the graph $G$. The basic properties of $con(G)$ and of its energy are established.

Characteristics of Common Neighborhood Graph under Graph Operations and on Cayley Graphs

Let G(V;E) be a graph. The common neighborhood graph (congraph) of G is a graph with vertex set V , in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of congraphs under graph operations; Graph :::::union:::::, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley graphs and its c...

On the Common–neighborhood Energy of a Graph

On common neighborhood graphs II

Let G be a simple graph with vertex set V (G). The common neighborhood graph or congraph of G, denoted by con(G), is a graph with vertex set V (G), in which two vertices are adjacent if and only if they have at least one common neighbor in G. We compute the congraphs of some composite graphs. Using these results, the congraphs of several special graphs are determined.

Graph Cospectrality using Neighborhood Matrices

In this note we address the problem of graph isomorphism by means of eigenvalue spectra of different matrix representations: the neighborhood matrix M̂ , its corresponding signless Laplacian QM̂ , and the set of higher order adjacency matrices M`s. We find that, in relation to graphs with at most 10 vertices, QM̂ leads to better results than the signless Laplacian Q; besides, when combined with M̂ ...

Neighborhood transformations on graph automata

We consider simulations of graph automata. We introduce two local transformations on the neighborhood: splitting and merging. We explain how to use such transformations, and their consequences on the topology of the simulated graph, the speed of the simulation and the memory size of simulating automata in some cases. As an example , we apply these transformations to graph automata embedded on s...