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...

Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra text] defined in Hartglass–Penneys’ article on canonical text]-algebras associated to planar algebra. Under mild conditions, is non-nuclear simple text]-algebra unique tracial state. There polynomial subalgebra together Dirac number operator such that spectral triple. We prove Haagerup-type bound of...

We prove that if G is a quasi-line graph with ∆(G) > ω(G) and ∆(G) ≥ 69, then χOL(G) ≤ ∆(G) − 1. Together with our previous work, this implies that if G is a claw-free graph with ∆(G) > ω(G) and ∆(G) ≥ 69, then χl(G) ≤ ∆(G)− 1.

Let G be a claw-free graph with order n and minimum degree δ. We improve results of Faudree et al. and Gould & Jacobson, and solve two open problems by proving the following two results. If δ = 4, then G has a 2-factor with at most (5n− 14)/18 components, unless G belongs to a finite class of exceptional graphs. If δ ≥ 5, then G has a 2-factor with at most (n− 3)/(δ − 1) components, unless G is...