The neighborhood polynomial of graph $G$ is the generating function for number vertex subsets which vertices have a common neighbor in $G$. In this paper, we investigate behavior under several operations. Specifically, provide an explicit formula obtained from given by attachment. We use result to propose recursive algorithm calculation polynomial. Finally, prove that can be found polynomial-ti...

a watching system in a graph $g=(v, e)$ is a set $w={omega_{1}, omega_{2}, cdots, omega_{k}}$, where $omega_{i}=(v_{i}, z_{i}), v_{i}in v$ and $z_{i}$ is a subset of closed neighborhood of $v_{i}$ such that the sets $l_{w}(v)={omega_{i}: vin omega_{i}}$ are non-empty and distinct, for any $vin v$. in this paper, we study the watching systems of line graph $k_{n}$ which is called triangular grap...

In 2022, the idea of a bivariate polynomial which represents number complete subgraphs graph with corresponding common neighborhood systems has been introduced in [3]. present work, we extend this notion to more restricted case by considering maximal connected cliques given graph. Besides characterizing corona two graphs, establish clique resulting from graphs. Received: June 2, 2023Accepted: J...

A neighborhood graph, which represents the instances as vertices and their relations as weighted edges, is the basis of many semi-supervised and relational models for node labeling and link prediction. Most methods employ a sequential process to construct the neighborhood graph. This process often consists of generating a candidate graph, pruning the candidate graph to make a neighborhood graph...

For a graph G, the neighborhood complex N [G] is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well-known result of Lovász that if ‖N [G]‖ is k-connected, then the chromatic number of G is at least k + 3. We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of ...

Chang, MS., C.Y. Tang and R.C.T. Lee, Solving the Euclidean bottleneck biconnected edge subgraph problem by 2-relative neighborhood graphs, Discrete Applied Mathematics 39 (1992) 1-12. In this paper, we discuss the Euclidean bottleneck biconnected edge subgraph problem. We shall first define a k-relative neighborhood graph which is similar to the relative neighborhood graph first proposed by To...

Let G = (V,E) be a simple undirected graph. N(G) = (V,EN ) is the neighborhood graph of G, if and only if EN = {{a, b} ∣ a ∕= b ∧ ∃x ∈ V : {x, a} ∈ E ∧ {x, b} ∈ E}. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph Nk(G) := N(N(. . . N(G))) of G. So we det...