Search

In this paper, the nonnegative Q-matrix completion problem is studied. A real n × n matrix is a Q-matrix if for k ∊ {1, . . . , n}, the sum of all k × k principal minors is positive. A digraph D is said to have nonnegative Q-completion if every partial nonnegative Q-matrix specifying D can be completed to a nonnegative Q-matrix. For nonnegative Q-completion problem, necessary conditions and sufficient conditions for a digraph to have nonnegative Q-completion are obtained. Further, the digraphs of order at most four that have nonnegative Q-completion have been studied. [ABSTRACT FROM AUTHOR]

A real matrix is a -matrix if for the sum of all principal minors is positive. A digraph is said to have positive -completion if every partial positive -matrix specifying can be completed to a positive -matrix. In this paper, necessary conditions for a digraph to have positive -completion are obtained and sufficient conditions for a digraph to have positive -completion are provided. The digraphs of order at most 4 that include all loops and have positive -completion are characterized. Tournaments whose complements have positive -completion are singled out. Further, some comparisons between the -matrix and positive -matrix completion problems have been made. [ABSTRACT FROM AUTHOR]

A graph G is singular with nullity η(G), if zero is an eigenvalue of its adjacency matrix with multiplicity η(G). If η(G) = 1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the kernel eigenvector. The set of vertices which are not in the core is called the periphery of G. A graph G with nullity one is called a minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. In this article, we describe the structure of a singular unicyclic graph and single out the class of unicyclic graphs which are minimal configurations. [ABSTRACT FROM AUTHOR]