On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme

We establish the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results

On the diameter and zero forcing number of some graph classes in the Johnson, Grassmann and Hamming association scheme

We determine the diameter of generalized Grassmann graphs and the zero forcing number of some generalized Johnson graphs, generalized Grassmann graphs and the Hamming graphs. Our work extends several previously known results.</p

Maximum nullity and zero forcing of circulant graphs

The zero forcing number of a graph has been applied to communication complexity, electrical powergrid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of agraph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterizethe zero forcing number of various circulant graphs, including families of bipartite circulants, as well as allcubic circulants. We extend the de nition of the Möbius ladder to a type of torus product to obtain boundson the minimum rank and the maximum nullity on these products. We obtain equality for torus products byemploying orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined thesenumbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds forall circulant graphs

Problems in graph theory and partially ordered sets

This dissertation answers problems in three areas of combinatorics - processes on graphs, graph coloring, and antichains in a partially ordered set.First we consider Zero Forcing on graphs, an iterative infection process introduced by AIM Minimum Rank - Special Graphs Workgroup in 2008. The Zero Forcing process is a graph infection process obeying the following rules: a white vertex is turned black if it is the only white neighbor of some black vertex. The Zero Forcing Number of a graph is the minimum cardinality over all sets of black vertices such that, after a finite number of iterations, every vertex is black. We establish some results about the zero forcing number of certain graphs and provide a counter example of a conjecture of Gentner and Rautenbach. This chapter is joint with Gabor Meszaros, Antonio Girao, and Chapter 3 appears in Discrete Math, Vol. 341(4).In the second part, we consider problems in the area of Dynamic Coloring of graphs. Originally introduced by Montgomery in 2001, the r-dynamic chromatic number of a graph G is the least k such that V(G) is properly colored, and each vertex is adjacent to at least r different colors. In this coloring regime, we prove some bounds for graphs with lattice like structures, hypercubes, generalized intervals, and other graphs of interest. Next, we establish some of the first results in the area of r-dynamic coloring on random graphs. The work in this section is joint with Peter van Hintum.In the third part, we consider a question about the structure of the partially ordered set of all connected graphs. Let G be the set of all connected graphs on vertex set [n]. Define the partial ordering \u3c on G as follows: for G,H G let G \u3c H if E(G) E(H). The poset (G

The Inverse Eigenvalue Problem of a Graph

Historically, matrix theory and combinatorics have enjoyed a powerful, mutually beneficial relationship. Examples include: Perron-Frobenius theory describes the relationship between the combinatorial arrangement of the entries of a nonnegative matrix and the properties of its eigenvalues and eigenvectors (see [53, Chapter 8]). The theory of vibrations (e.g., of a system of masses connected by strings) provides many inverse problems (e.g., can the stiffness of the springs be prescribed to achieve a system with a given set of fundamental vibrations?) whose resolution intimately depends upon the families of matrices with a common graph (see [46, Chapter 7]). The Inverse Eigenvalue Problem of a graph (IEP-G), which is the focus of this chapter, is another such example of this relationship. The IEP-G is rooted in the 1960s work of Gantmacher, Krein, Parter and Fielder, but new concepts and techniques introduced in the last decade have advanced the subject significantly and catalyzed several mathematically rich lines of inquiry and application. We hope that this chapter will highlight these new ideas, while serving as a tutorial for those desiring to contribute to this expanding area

Novel Techniques for the Zero-Forcing and p-Median Graph Location Problems

This thesis presents new methods for solving two graph location problems, the p-Median problem and the zero-forcing problem. For the p-median problem, I present a branch decomposition based method that finds the best p-median solution that is limited to some input support graph. The algorithm can be used to either find an integral solution from a fractional linear programming solution, or it can be used to improve on the solutions given by a pool of heuristics. In either use, the algorithm compares favorably in running time or solution quality to state-of-the-art heuristics. For the zero-forcing problem, this thesis gives both theoretical and computational results. In the theoretical section, I show that the branchwidth of a graph is a lower bound on its zero-forcing number, and I introduce new bounds on the zero-forcing iteration index for cubic graphs. This thesis also introduces a special type of graph structure, a zero-forcing fort, that provides a powerful tool for the analysis and modeling of zero-forcing problems. In the computational section, I introduce multiple integer programming models for finding minimum zero-forcing sets and integer programming and combinatorial branch and bound methods for finding minimum connected zero-forcing sets. While the integer programming methods do not perform better than the best combinatorial method for the basic zero-forcing problem, they are easily adapted to the connected zero-forcing problem, and they are the best methods for the connected zero-forcing problem

A lower bound on the zero forcing number

In this note, we study a dynamic vertex coloring for a graph G. In particular, one starts with a certain set of vertices black, and all other vertices white. Then, at each time step, a black vertex with exactly one white neighbor forces its white neighbor to become black. The initial set of black vertices is called a zero forcing set if by iterating this process, all of the vertices in G become black. The zero forcing number of G is the minimum cardinality of a zero forcing set in G, and is denoted by Z(G). Davila and Kenter have conjectured in 2015 that Z(G)≥(g−3)(δ−2)+δ where g and δ denote the girth and the minimum degree of G, respectively. This conjecture has been proven for graphs with girth g≤10. In this note, we present a proof for g≥5, δ≥2, thereby settling the conjecture