8 citations · 26 across the 23 of their papers we have counts for
25 papers · 1 filter
On the Number of Zero Forcing Minimal Forts on Trees
Nguyen Hoang Dat, Franklin H. J. Kenter
We solve a conjecture by Becker et al. (arXiv:2404.05963) on the topic of zero forcing regarding the number of minimal forts of a tree. They conjectured and we prove $\mathcal{F}_{…
Leaky Forcing: Extending Zero Forcing Results to a Fault-Tolerant Setting
Beth Bjorkman, Lei Cao, Franklin Kenter +3
We study a recent variation of zero forcing called leaky forcing. Zero forcing is a propagation process on a network whereby some nodes are initially blue with all others white. Bl…
Forts, (fractional) zero forcing, and Cartesian products of graphs
Thomas R. Cameron, Leslie Hogben, Franklin H. J. Kenter +2
The (disjoint) fort number and fractional zero forcing number are introduced and related to existing parameters including the (standard) zero forcing number. The fort hypergraph is…
Zero Forcing on 2-connected Outerplanar Graphs
Nolan Ison, Mark Kempton, Franklin Kenter
We determine upper and lower bounds on the zero forcing number of 2-connected outerplanar graphs in terms of the structure of the weak dual. We show that the upper bound is always…
Improved bounds on the cop number when forbidding a minor
Franklin Kenter, Erin Meger, Jérémie Turcotte
Andreae (1986) proved that the cop number of connected -minor-free graphs is bounded for every graph . In particular, the cop number is at most if contains n…
Sparsity of Graphs that Allow Two Distinct Eigenvalues
Wayne Barrett, Shaun Fallat, Veronika Furst +5
The parameter of a graph is the minimum number of distinct eigenvalues over the family of symmetric matrices described by . It is shown that the minimum number of edg…
A Geometric Chung Lu model and the Drosophila Medulla connectome
Susama Agarwala, Franklin Kenter
Many real world graphs have edges correlated to the distance between them, but, in an inhomogeneous manner. While the Chung-Lu model and the geometric random graph models both are…
A Tale of Two Limits: An Extremal Pagerank Problem
Joseph Farnan, Franklin H. J. Kenter
For a directed graph, the Pagerank algorithm emulates a random walker on the graph that occasionally "jumps" to a random vertex based on a jumping parameter . Upon completion, t…
A zero forcing technique for bounding sums of eigenvalue multiplicities
Franklin H. j. Kenter, Jephian C. -H. Lin
Given a graph , one may ask: "What sets of eigenvalues are possible over all weighted adjacency matrices of ?" (The weight of an edge is positive or negative, while the diago…
An upper bound for the -power domination number in -uniform hypergraphs
Joseph S. Alameda, Franklin Kenter, Karen Meagher +1
Generalizing work on graphs, Chang and Roussel introduced -power domination in hypergraphs and conjectured the upper bound for the -power domination number for -uniform hy…
Leaky Forcing: A New Variation of Zero Forcing
Shannon Dillman, Franklin Kenter
Zero forcing is a one-player game played on a graph. The player chooses some set of vertices to color, then iteratively applies a color change rule: If all but one of a colored ver…
Computing Bounds on Product-Graph Pebbling Numbers
Franklin Kenter, Daphne Skipper, Dan Wilson
Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling…
Pebbling on Directed Graphs with Fixed Diameter
John Asplund, Franklin Kenter
Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placin…
Pebbling on Graph Products and other Binary Graph Constructions
John Asplund, Glenn Hurlbert, Franklin Kenter
Pebbling on graphs is a two-player game which involves repeatedly moving a pebble from one vertex to another by removing another pebble from the first vertex. The pebbling number $…
On the error of a priori sampling: zero forcing sets and propagation time
Franklin H. J. Kenter, Jephian C. -H. Lin
Zero forcing is an iterative process on a graph used to bound the maximum nullity. The process begins with select vertices as colored, and the remaining vertices can become colored…
The relationship between -forcing and -power domination
Daniela Ferrero, Leslie Hogben, Franklin H. J. Kenter +1
Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In b…
Power propagation time and lower bounds for power domination number
Daniela Ferrero, Leslie Hogben, Franklin H. J. Kenter +1
We present a counterexample to a lower bound for the power domination number given in Liao, Power domination with bounded time constraints, J. Comb. Optim. 31 (2016)725-742. We als…
On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs
Keivan Hassani Monfared, Paul Horn, Franklin H. J. Kenter +3
The principal permanent rank characteristic sequence is a binary sequence where if there exists a principal square submatrix of size with nonzero…
On the distance spectra of graphs
Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy +8
The distance matrix of a graph is the matrix containing the pairwise distances between vertices. The distance eigenvalues of are the eigenvalues of its distance matrix and…
Lower Bounds on the Distance Domination Number of a Graph
Randy Davila, Caleb Fast, Michael Henning +1
For an integer , a (distance) -dominating set of a connected graph is a set of vertices of such that every vertex of is at distance at mo…
Proof of a conjecture of Graham and Lovász concerning unimodality of coefficients of the distance characteristic polynomial of a tree
Ghodratollah Aalipour, Aida Abiad, Zhanar Berikkyzy +4
We establish a conjecture of Graham and Lovász that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal; we also prove they are log-conca…
A linear k-fold Cheeger inequality
Franklin Kenter, Mary Radcliffe
Given an undirected graph , the classical Cheeger constant, , measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon…
Necessary Spectral Conditions for Coloring Hypergraphs
Franklin H. J. Kenter
Hoffman proved that for a simple graph , the chromatic number obeys where and are the maximal and minimal eigenvalues of the…
A Linear Cheeger Inequality using Eigenvector Norms
Franklin H. J. Kenter
The Cheeger constant, , is a measure of expansion within a graph. The classical Cheeger Inequality states: where is the first nontri…