Edges Of Qn Research Articles

Kirchhoff Indices and Numbers of Spanning Trees of Molecular Graphs Derived from Linear Crossed Polyomino Chain

Polyomino systems are widely studied in organic chemistry, especially in polycyclic aromatic compounds. Let Qn be a linear crossed polyomino chain with n complete quadrangles. In this article, we construct a family of molecular graphs obtained by randomly deleting some specified edges of Qn . By using the decomposition theorem of the Laplacian matrix characteristic polynomial, explicit formulas of Kirchhoff indices and numbers of spanning trees of those molecular graphs derived from Qn are obtained. Moreover, their Kirchhoff indices are shown to be almost one quarter of their Wiener indices.

EDGE-BIPANCYCLICITY OF HYPERCUBES WITH CONDITIONAL FAULTS

In this paper, we consider the conditionally faulty graphs G that each vertex of G is incident with at least m fault-free edges, 2 ≤ m ≤ n - 1. We extend the limitation m ≥ 2 in all previous results of edge-bipancyclicity with faulty edges and faulty vertices. Let fe (respectively, fv) denotes the number of faulty edges (respectively, faulty vertices) in an n-dimensional hypercube Qn. For all m, we show that every fault-free edge of Qn lies on a fault-free cycle of every even length from 4 to |V| - 2fv inclusive provided fe + fv ≤ n - 2. This result is not only optimal, but also improves on the previously best known results reported in the literature.

Poisson Convergence in the n‐Cube

AbstractWe consider two types of random subgraphs of the n‐cube Qn obtained by independent deletion the vertices (together with all edges incident with them) or the edges of Qn, respectively, with a prescribed probability q = 1 — p. For these two probabilistic models we determine some values of the probability p for which the number of (isolated) k‐dimensional subcubes or the number of vertices of a given degree k, respectively, has asymptotically a Poisson or a Normal distribution. The technique which will be used is that of Poisson convergence introduced by BARBOUR [1] (see also [4]).