Recognizing generalized Sierpiński graphs

Let H be an arbitrary graph with vertex set V (H) = [nH] = {l,?, nH}. The generalized Sierpi?ski graph SnH , n ? N, is defined on the vertex set [nH]n, two different vertices u = un ?u1 and v = vn ? v1 being adjacent if there exists an h? [n] such that (a) ut = vt, for t > h, (b) uh ? vh and uhvh ? E(H), and (c) ut = vh and vt = uh for t < h. If H is the complete graph Kk, then we speak of the Sierpi?ski graph Sn k . We present an algorithm that recognizes Sierpi?ski graphs Sn k in O(|V (Sn k )|1+1=n) = O(|E(Sn k )|) time. For generalized Sierpi?ski graphs SnH we present a polynomial time algorithm for the case when H belong to a certain well defined class of graphs. We also describe how to derive the base graph H from an arbitrarily given SnH .

Adjacency and Laplacian spectra of variants of neighborhood corona of graphs constrained by vertex subsets

In this paper, we define some variants of corona of graphs namely, subdivision (respectively, [Formula: see text]-graph, [Formula: see text]-graph, total) neighborhood corona, [Formula: see text]-graph (respectively, [Formula: see text]-graph, total) semi-edge neighborhood corona, [Formula: see text]-graph (respectively, total) semi-vertex neighborhood corona of graphs constrained by vertex subsets. These corona operations generalize some existing corona operations such as subdivision ([Formula: see text]-graph, [Formula: see text]-graph, total) double neighborhood corona, subdivision vertex (respectively, edge) neighborhood corona, [Formula: see text]-graph vertex (respectively, edge) neighborhood corona of graphs. First, we consider a matrix in specific form and determine its spectrum. Then by using this, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the new graphs when the base graph is regular. Also, we deduce the characteristic polynomials of the adjacency and Laplacian matrices of the above mentioned particular cases from our results.

Regularity of bicyclic graphs and their powers

Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.

Morphological Analysis by Surface Patterns and by Graph

In this article, we propose a comparison between our two morphological analyzers, which we have developed in recent years. The first is based on surface patterns Arabic words, the second is an analyzer which combines Buckwalter approach and the approach of morphological analysis in base graph. The comparison is made on a corpus of 1400 Arabic words that generalize all cases of Arabic derived words. The results obtained show the interest and the advantages of each analyzer.

On distances in generalized Sierpiński graphs

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms of the distance between vertices of the base graph G. In particular, we deduce a recursive formula for the distance between an arbitrary vertex and an extreme vertex of S(G, t), and we obtain a recursive formula for the distance between two arbitrary vertices of S(G, t) when the base graph is triangle-free. From these recursive formulas, we provide algorithms to compute the distance between vertices of S(G, t). In addition, we give an explicit formula for the diameter and radius of S(G, t) when the base graph is a tree.

Small Subgraphs in the Trace of a Random Walk

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.

On the roman domination number of generalized Sierpiński graphs

A map f : V?(0,1,2) is a Roman dominating function on a graph G = (V,E) if for every vertex v ? V with f(v)=0, there exists a vertex u, adjacent to v, such that f(u)=2. The weight of a Roman dominating function is given by f(V)=?u?V f(u). The minimum weight among all Roman dominating functions on G is called the Roman domination number of G. In this article we study the Roman domination number of Generalized Sierpi?ski graphs S(G,t). More precisely, we obtain a general upper bound on the Roman domination number of S(G,t) and discuss the tightness of this bound. In particular, we focus on the cases in which the base graph G is a path, a cycle, a complete graph or a graph having exactly one universal vertex.