Inverses of triangular matrices and bipartite graphs

Some lower bounds for the $L$-intersection number of graphs

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

On Inverses of Triangular Matrices with Monotone Entries

This note employs recurrence techniques to obtain entry-wise optimal inequalities for inverses of triangular matrices whose entries satisfy some monotonicity constraints. The derived bounds are easily computable.

Inverses of bicyclic graphs

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contract...

Group inverses of matrices with path graphs

A simple formula for the group inverse of a 2× 2 block matrix with a bipartite digraph is given in terms of the block matrices. This formula is used to give a graph-theoretic description of the group inverse of an irreducible tridiagonal matrix of odd order with zero diagonal (which is singular). Relations between the zero/nonzero structures of the group inverse and the Moore-Penrose inverse of...

Some Observations on Inverses of Band Matrices and Low Rank Perturbations of Triangular Matrices