27,881 research outputs found

    Extrema of graph eigenvalues

    Full text link
    In 1993 Hong asked what are the best bounds on the kk'th largest eigenvalue λk(G)\lambda_{k}(G) of a graph GG of order nn. This challenging question has never been tackled for any 2<k<n2<k<n. In the present paper tight bounds are obtained for all k>2,k>2, and even tighter bounds are obtained for the kk'th largest singular value λk∗(G).\lambda_{k}^{\ast}(G). Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large can λk(G)+λk(Gˉ)\lambda_{k}(G)+\lambda_{k}(\bar{G}) be?? These constructions are successful also in another open question: How large can the Ky Fan norm λ1∗(G)+...+λk∗(G)\lambda_{1}^{\ast}(G)+...+\lambda_{k}^{\ast }(G) be ?? Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs. In the final section, several results and problems are restated for (−1,1)(-1,1)-matrices, which seem to provide a more natural ground for such research than graphs. Many of the results in the paper are paired with open questions and problems for further study.Comment: 32 page

    Testing of random matrices

    Get PDF
    Let nn be a positive integer and X=[xij]1≤i,j≤nX = [x_{ij}]_{1 \leq i, j \leq n} be an n×nn \times n\linebreak \noindent sized matrix of independent random variables having joint uniform distribution \hbox{Pr} {x_{ij} = k \hbox{for} 1 \leq k \leq n} = \frac{1}{n} \quad (1 \leq i, j \leq n) \koz. A realization M=[mij]\mathcal{M} = [m_{ij}] of XX is called \textit{good}, if its each row and each column contains a permutation of the numbers 1,2,...,n1, 2,..., n. We present and analyse four typical algorithms which decide whether a given realization is good
    • …
    corecore