Let forb(m,F ) denote the maximum number of columns possible in a (0,1)matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F . We consider cases where the configuration F has a number of columns that grows with m. For a k× ` matrix G, define s ·G to be the concatenation of s copies of G. In a number of cases we determine forb(m,mα ·G) is Θ(mk+α). ...

It is well known that the inverse C = c i;j ] of an irreducible nonsingular symmetric tridiagonal matrix is a Green matrix, i.e. it satisses c i;j = u i v j for i j and two sequences of real numbers fu i g and fv i g. A similar result holds for nonsymmetric matrices A. There the inverse is given by four sequences fu i g; fv i g; fy i g; and fy i g: Here we characterize certain properties of A i...

As is well known, the fundamental matrix (I − P + eπ) plays an important role in the performance analysis of Markov systems, where P is the transition probability matrix, e is the column vector of ones, and π is the row vector of the steady state distribution. It is used to compute the performance potential (relative value function) of Markov decision processes under the average criterion, such...

We show that the Hamiltonicity of a regular graph G can be fully characterized by the numbers of blocks of consecutive ones in the binary matrix A+ I, where A is the adjacency matrix of G, I is the unit matrix, and the blocks can be either linear or circular. Concretely, a k-regular graph G with girth g(G) 5 has a Hamiltonian circuit if and only if the matrix A + I can be permuted on rows such ...

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,−1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, −1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in simil...

Let ∆n denote the set of n×n matrices of non-negative integers which have each row and column sum equal to k. Let Λn denote the subset of all binary matrices (matrices of zeroes and ones) in ∆n. If G is a bipartite multigraph let B(G) denote the usual ‘biadjacency’ matrix of G. That is, B(G) is the matrix with rows and columns respectively corresponding to the vertices in the two parts of G, an...

2.1.1 matrix games A 2-player zero-sum game, or a matrix game, is defined by a matrix A ∈ Rm×n, called the payoff matrix. There are two players with opposing interests: the row player (minimizer) and the column player (maximizer). At each step of the play, the row player selects a row i, and the column player selects a column j, then the row player pays to the column player the value aij of the...

Let G be a subgraph of a complete bipartite graph Kn,n. Let N(G) be a 0-1 incidence matrix with edges of Kn,n against images of G under the automorphism group of Kn,n. A diagonal form of N(G) is found for every G, and whether the row space of N(G) over Zp contains the vector of all 1’s is determined. This re-proves Caro and Yuster’s results on zero-sum bipartite Ramsey numbers [3], and provides...

The PageRank is a widely used scoring function of networks in general and of the World Wide Web graph in particular. The PageRank is defined for directed graphs, but in some special cases applications for undirected graphs occur. In the literature it is widely noted that the PageRank for undirected graphs are proportional to the degrees of the vertices of the graph. We prove that statement for ...

Graham et al. [3] proved a very attractive theorem about the determinant of the distance matrix DG of a connected graph G as a function of the distance matrix of its 2-connected blocks. In a connected graph, the distance between two vertices d(u, v) is the length of the shortest path between them. Let A be an n n matrix. Recall that for 1 i, j n, the cofactor ci,j is defined as ( 1) iþj times t...