Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph (see [AIM Minimum Rank - Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler, S. Cioab$\breve{\text{a}}$, D. Cvetkovi$\acute{\text{c}}$, S. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi$\acute{\text{c}}$, H. van der Holst, K. Vander Meulen, and A. Wangsness). Linear Algebra Appl., 428(7):1628--1648, 2008]). In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters that give lower bounds on the maximum nullity of a graph. Inparticular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdiére type parameter and vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.

Leap Zagreb Connection Numbers for Some Networks Models

The main object of this study is to determine the exact values of the topological indices which play a vital role in studying chemical information, structure properties like QSAR and QSPR. The first Zagreb index and second Zagreb index are among the most studied topological indices. We now consider analogous graph invariants, based on the second degrees of vertices, called leap Zagreb indices. We compute these indices for Tickysim SpiNNaker model, cyclic octahedral structure, Aztec diamond and extended Aztec diamond.

Symmetric matrices, Catalan paths, and correlations

International audience Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.

Spanning Trees of the Generalised Union Jack Lattice

The Union Jack lattice UJL(n, m) with toroidal boundary condition can be obtained from an n×m square lattice with toroidal boundary condition by inserting a new vertex vf to each face f and adding four edges (vf, ui(f)), where u1(f), u2(f), u3(f), and u4(f) are four vertices on the boundary of f. The Union Jack lattice has been studied extensively by statistical physicists. In this article, we consider the problem of enumeration of spanning trees of the so-called generalised Union Jack lattice UDn, which is obtained from the Aztec diamond $AD_n^t$ of order n with toroidal boundary condition by inserting a new vertex vf to each face f and adding four edges (vf, ui(f)), where u1(f), u2(f), u3(f) and u4(f) are four vertices on the boundary of f.