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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 $I \cap J$ 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 $ℒ (t) = {x : f (x) \geq t}$ . 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 graphic generalization of arithmetic.

Khan, Bilal, Bhutani, Kiran R., Kahrobaei, Delaram (2007)

Integers

A graphical way to solve the Boolean matrix equations $A X = B$ and $X A = B$

U. J. Nieminen (1974)

Kybernetika

A graph-theoretic method for choosing a spanning set for a finite-dimensional vector space, with applications to the Grossman-Larson-Wright module and the Jacobian conjecture.

Singer, Dan (2009)

The Electronic Journal of Combinatorics [electronic only]

A graph-theoretical representation of PL-manifolds - A survey on crystallizations.

M. Ferri, C. Gagliardi, L. Grasselli (1986)

Aequationes mathematicae

A Group of Automorphisms of the Rooted Dyadic Tree and Associated Gelfand Pairs

Daniele D’Angeli, Alfredo Donno (2009)

Rendiconti del Seminario Matematico della Università di Padova

A Hamiltonian cycle and a $1$ -factor in the fourth power of a graph

Elena Wisztová (1985)

Časopis pro pěstování matematiky

A Hamiltonian property of connected sets in the alternative set theory.

Zlatoš, P. (1993)

Acta Mathematica Universitatis Comenianae. New Series

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 $K_{r} + \bar{K_{s}}$ 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 $[k, k + 1]$ -factor containing a given Hamiltonian cycle.

Cai, Maocheng, Li, Yanjun (1999)

The Electronic Journal of Combinatorics [electronic only]

A Kotzig type theorem for non-orientable surfaces

Stanislav Jendroľ, Milan Tuhársky (2006)

Mathematica Slovaca

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