We consider the problem of structure prediction for sparse LU factorization with partial pivoting. In this context, it is well known that the column elimination tree plays an important role for matrices satisfying an irreducibility condition, called the strong Hall property. Our primary goal in this paper is to address the structure prediction problem for matrices satisfying a weaker assumption...

The eccentricity matrix E(G) of a connected graph G is obtained from the distance by keeping largest nonzero entries in each row and column, leaving zeros remaining ones. eigenvalues are E-eigenvalues G. In this article, we find inertia matrices trees. Interestingly, any tree on more than 4 vertices with odd diameter has exactly two positive negative (irrespective structure tree). Also, show th...

Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix F ∈ F (as a row and column permutation of F ). A matrix M without repeated columns is F-saturated if M is F-admissible but t...

Matrices of dimensions m × 1 and 1 × n are called column and row vectors, respectively. We will typically denote column and row vectors by lower case Latin letters, e.g. a, b, x, y and other matrices by upper case Latin letters, e.g. A, B, X, Y . The scalars (or 1 × 1 matrices) will be frequently denoted by Greek letters α, β, λ, μ, etc. Unless stated otherwise, all scalars will be real numbers...

This paper studies the set of n × n matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is isomorphic to the set of arbitrary (n−1)×(n−1) matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower...

Recently, Jeĺınek derived that the number of self-dual interval orders of reduced size n is twice the number of row-Fishburn matrices of size n by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of...

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance $G$ keeping largest nonzero entries in each row and column leaving zeros remaining ones. $\mathcal{E}$-eigenvalues are eigenvalues $\mathcal{E}(G)$. modulus an eigenvalue $\mathcal{E}$-spectral radius $G$. $\mathcal{E}$-energy sum absolute values all In this article, we study some extrem...