On the binding number of some Hallian graphs
For any graph , let and denote the vertex set and the edge set of respectively. The Boolean function graph of is a graph with vertex set and two vertices in are adjacent if and only if they correspond to two adjacent vertices of , two adjacent edges of or to a vertex and an edge not incident to it in . For brevity, this graph is denoted by . In this paper, structural properties of and its complement including traversability and eccentricity properties are studied. In addition,...
We present some extensions of the Borel-Cantelli Lemma in terms of moments. Our result can be viewed as a new improvement to the Borel-Cantelli Lemma. Our proofs are based on the expansion of moments of some partial sums by using Stirling numbers. We also give a comment concerning the results of Petrov V.V., A generalization of the Borel-Cantelli Lemma, Statist. Probab. Lett. 67 (2004), no. 3, 233–239.
Let be the algebraic connectivity, and let be the Laplacian spectral radius of a -connected graph with vertices and edges. In this paper, we prove that with equality if and only if is the complete graph or . Moreover, if is non-regular, then where stands for the maximum degree of . Remark that in some cases, these two inequalities improve some previously known results.
We use an algebraic method to classify the generalized permutation star-graphs, and we use the classification to determine the toughness of all generalized permutation star-graphs.
A -simplex is the convex hull of affinely independent vertices of the unit -cube . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute -simplices in can be represented by -matrices of size whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part of the transposed inverse of is doubly stochastic and has the same support...
Homotopic methods are employed for the characterization of the complete digraphs which are the composition of non-trivial highly regular tournaments.