In this paper, we show that if G is an l-connected claw-free graph with minimum degree at least three and l ∈ {2, 3}, then for any maximum independent set S, there exists a 2-factor in which each cycle contains at least l − 1 vertices in S.

Turán's famous tetrahedron problem is to compute the Turán density of K 4 3 $K_4^3$ . This equivalent determining maximum ℓ 1 $\ell _1$ -norm codegree vector a -free n $n$ -vertex 3-uniform hypergraph. We introduce new way for measuring extremality hypergraphs and determine asymptotically extremal function in our notion. The squared sum, co 2 ( G ) $\mbox{co}_2(G)$ , hypergraph $G$ sum codegree...

Every n-vertex graph has two vertices with the same degree (if n ≥ 2). In general, let rep(G) be the maximum multiplicity of a vertex degree in G. An easy counting argument yields rep(G) ≥ n/(2d − 2s + 1), where d is the average degree and s is the minimum degree of G. Equality can hold when 2d is an integer, and the bound is approximately sharp in general, even when G is restricted to be a tre...

The directed power graph G?(G) of a group G is the simple digraph with vertex set such that x?y if y x. G, denoted by G(G), underlying graph. In this paper, for groups and H, following proved. If has no quasicyclic subgroup Cp? which trivial intersection every cyclic K K??Cp?, then G(G)?G(H) implies G?(G)?G?(H). particular, any two torsion-free having isomorphic graphs have graphs.

A graph G is triangularly connected if for every pair of edges e1, e2 ∈ E(G), G has a sequence of 3-cycles C1, C2, · · · , Cl such that e1 ∈ C1, e2 ∈ Cl and such that E(Ci) ∩ E(Ci+1) 6= ∅, (1 ≤ i ≤ l − 1). In this paper it is shown that every triangularly connected claw-free graph G with |E(G)| ≥ 3 is vertex pancyclic. This implies the former results in [2], [3], [4] and [5] that every connecte...

This paper proposes an original approach to cluster multi-component data sets, including an estimation of the number of clusters. From the construction of a minimal spanning tree with Prim’s algorithm, and the assumption that the vertices are approximately distributed according to a Poisson distribution, the number of clusters is estimated by thresholding the Prim’s trajectory. The correspondin...