On the powers of tournament matricesVarious results are derived about the adjacency matrix of a tournament; in particular, it is shown that if a tournament T n is irreducible and n>-5, then its matrix is primit
Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination ...
adjacency matrixboolean matrixcomputational complexitygraph diameterrandom graphsSummary: In this paper, a fast algorithm is proposed to calculate $k ^{th }$ power of an $n\times n$ Boolean matrix that requires $O(kn ^{3} p)$ addition operations, where $p$ is the probability that an ...
Adjacency MatrixGraph DiameterComputational ComplexityIn this paper, a fast algorithm is proposed to calculate k{sup}(th) power of an n脳n Boolean matrix that requires O(kn{sup}3p) addition operations, where p is the probability that an entry of the matrix is 1. The algorithm generates a ...
For example, let A,λ,d⃗, and δ be the adjacency matrix of G, the largest eigenvalue of A, the degree vector of G, and the minimum degree of G, respectively. Since A is a symmetric matrix, it is well known that the proposition 2mδ2≤2∑uv∈E(G)dG(u)dG(v)=d⃗TAd⃗...
Adjacency matrixLaplacian matrixThis note deals with the relationship between the total number of k -walks in a graph, and the sum of the k -th powers of its vertex degrees. In particular, it is shown that the the number of all k -walks is upper bounded by the sum of the k -th ...
The Laplacian matrix of G is L(G) = D(G) A(G), where D(G) is the diagonal matrix of its vertex degrees and A(G) is the adjacency matrix. Let mu(1) >= mu(2) >= ... >= mu(n-1) >= mu(n) = 0 be the Laplacian eigenvalues of G. For a graph G and a real number...