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A graph associated to proper non-small ideals of a commutative ring

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring R , denoted by G ( R ) , is a graph with all non-small proper ideals of R as vertices and two distinct vertices I and J are adjacent if and only if I ∩ J is not small in R . In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

A graph-based estimator of the number of clusters

Gérard Biau, Benoît Cadre, Bruno Pelletier (2007)

ESAIM: Probability and Statistics

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given n independent observations X1,...,Xn drawn from an unknown multivariate probability density f, we propose a new approach to estimate the number of connected components, or clusters, of the t-level set ℒ ( t ) = { x : f ( x ) ≥ t } . The basic idea is to form a rough skeleton of the set ℒ ( t ) using any preliminary estimator of f, and to count the number of connected components of the resulting graph. Under...

A Havel-Hakimi type procedure and a sufficient condition for a sequence to be potentially S r , s -graphic

Jian Hua Yin (2012)

Czechoslovak Mathematical Journal

The split graph K r + K s ¯ on r + s vertices is denoted by S r , s . A non-increasing sequence π = ( d 1 , d 2 , ... , d n ) of nonnegative integers is said to be potentially S r , s -graphic if there exists a realization of π containing S r , s as a subgraph. In this paper, we obtain a Havel-Hakimi type procedure and a simple sufficient condition for π to be potentially S r , s -graphic. They are extensions of two theorems due to A. R. Rao (The clique number of a graph with given degree sequence, Graph Theory, Proc. Symp., Calcutta 1976, ISI Lect. Notes Series...

A linear algorithm for the two paths problem on permutation graphs

C.P. Gopalakrishnan, C. Pandu Rangan (1995)

Discussiones Mathematicae Graph Theory

The 'two paths problem' is stated as follows. Given an undirected graph G = (V,E) and vertices s₁,t₁;s₂,t₂, the problem is to determine whether or not G admits two vertex-disjoint paths P₁ and P₂ connecting s₁ with t₁ and s₂ with t₂ respectively. In this paper we give a linear (O(|V|+ |E|)) algorithm to solve the above problem on a permutation graph.

A Linear Time Algorithm for Computing Longest Paths in Cactus Graphs

Markov, Minko, Ionut Andreica, Mugurel, Manev, Krassimir, Tapus, Nicolae (2012)

Serdica Journal of Computing

ACM Computing Classification System (1998): G.2.2.We propose an algorithm that computes the length of a longest path in a cactus graph. Our algorithm can easily be modified to output a longest path as well or to solve the problem on cacti with edge or vertex weights. The algorithm works on rooted cacti and assigns to each vertex a two-number label, the first number being the desired parameter of the subcactus rooted at that vertex. The algorithm applies the divide-and-conquer approach and computes...

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