arXiv subjects

Joel Jeffries

Publications and source records attributed to Joel Jeffries.

6 recordsLinked to original sources

arXiv2026

Small $q$-kernels in digraphs with minimum in-degree $δ$

For a digraph $D$, a subset $Q\subseteq V(D)$ is called a $q$-kernel if $Q$ is an independent set and all vertices in $V(D)$ are reachable from $Q$ via a directed path of length at most $q$. Given integers $q\geq 2$ and $δ\geq 1$, Spiro arXiv:2404.07305 [math.CO] posed the question: what is the smallest constant $c_{δ,q}$ such that every digraph $D$ with minimum in-degree $δ$ has a $q$-kernel of size at most $c_{δ,q}|V(D)|$? We show the constants $c_{δ,q}$ are monotone in both $δ$ and $q$, and we improve upon the known upper bounds for $c_{δ,q}$. Our main results show $\frac{1}{δ+1} \leq c_{δ,q}\leq \frac{1}{\lfloor\sqrt{δ+1}\rfloor+1}$ for all $q \geq 3$ and $δ\geq 1$, and $ c_{δ,q}=\frac{1}{δ+1}$ whenever $δ\geq 1$ and $q \geq \left\lceil\frac{3δ}{2}\right\rceil + 1$.

math.CO↗

arXiv2026

Schuttes property for sets of tournaments and an application to dice games

A tournament has Schuttes property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set. In 1963, Erdos provided bounds for $f(k)$, the smallest order of an $S_k$ tournament. Schuttes property has various applications, including the design of unfair dice games. A set of dice introduced by James Grime motivates a generalization of Schuttes property to sets of tournaments: a set of tournaments on the same vertex set has property $S_k$ if for every set of $k$ vertices, there is a vertex which dominates the set in at least one of the tournaments. We explore this generalization and provide bounds on the fewest number of vertices needed to have an $S_k$ set of $m$ tournaments. We then apply these results to introduce a few new sets of dice similar to Grimes dice that can be used to play a game that gives one player an advantage.

math.CO↗

arXiv2026

Enumerating Prime Patterns in Juggling Variations

Juggling patterns can be mathematically modeled as closed walks within directed state graphs. In this paper, we present a unified framework of unbounded juggling patterns and its variations (including multiplex, colored, and passing) primarily through the formalism of the juggling state. By extending this state-based approach and utilizing combinatorial tools such as set partitions and filled Ferrers diagrams, we find and prove a new lower bound on the number of $b$-ball prime patterns with period $n$. Further, we determine exact counts for 2-ball multiplex, 1-ball passing, and 2-ball colored juggling patterns, as well as a lower bound for 2-ball passing. We also provide an extensive analysis of the asymptotic growth rates for these pattern counts. Finally, we formalize the infinite state graph, $G_\infty$, and utilize flip-reverse involutions to establish bijections between classes of prime patterns, exploring how fixing a specific state influences the enumeration of prime walks.

math.CO↗

arXiv2023

On the Edge Derivative of the Normalized Laplacian with Applications to Kemeny's Constant

In a connected graph, Kemeny's constant gives the expected time of a random walk from an arbitrary vertex $x$ to reach a randomly-chosen vertex $y$. Because of this, Kemeny's constant can be interpreted as a measure of how well a graph is connected. It is generally unknown how the addition or removal of edges affects Kemeny's constant. Inspired by the directional derivative of the normalized Laplacian, we derive the directional derivative of Kemeny's constant for several graph families. In addition, we find sharp bounds for the directional derivative of an eigenvalue of the normalized Laplacian and bounds for the directional derivative of Kemeny's constant.

math.CO↗

arXiv2022

Complements of coalescing sets

We consider matrices of the form $qD+A$, with $D$ being the diagonal matrix of degrees, $A$ being the adjacency matrix, and $q$ a fixed value. Given a graph $H$ and $B\subseteq V(G)$, which we call a coalescent pair $(H,B)$, we derive a formula for the characteristic polynomial where a copy of same rooted graph $G$ is attached by the root to \emph{each} vertex of $B$. Moreover, we establish if $(H_1,B_1)$ and $(H_2,B_2)$ are two coalescent pairs which are cospectral for any possible rooted graph $G$, then $(H_1,V(H_1)\setminus B_1)$ and $(H_2,V(H_2)\setminus B_2)$ will also always be cospectral for any possible rooted graph $G$.

math.CO↗

arXiv2022

Spectral faux trees

A spectral faux tree with respect to a given matrix is a graph which is not a tree but is cospectral with a tree for the given matrix. We consider the existence of spectral faux trees for several matrices, with emphasis on constructions. For the Laplacian matrix, there are no spectral faux trees. For the adjacency matrix, almost all trees are cospectral with a faux tree. For the signless Laplacian matrix, spectral faux trees can only exist when the number of vertices is of the form $n=4k$. For the normalized adjacency, spectral faux trees exist when the number of vertices $n\ge 4$, and we give an explicit construction for a family whose size grows exponentially with $k$ for $n=αk+1$ where $α$ is fixed.

math.CO↗