Lucas Mol

Lucas Mol
Thompson Rivers University · Department of Mathematics and Statistics

Doctor of Philosophy

About

42
Publications
2,639
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168
Citations
Citations since 2017
40 Research Items
167 Citations
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2017201820192020202120222023010203040
2017201820192020202120222023010203040
2017201820192020202120222023010203040
Introduction
I am an Assistant Teaching Professor at Thompson Rivers University in Kamloops, BC, Canada. My research interests are in combinatorics on words and graph theory.

Publications

Publications (42)
Preprint
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We study various aspects of Dyck words appearing in binary sequences, where $0$ is treated as a left parenthesis and $1$ as a right parenthesis. We show that binary words that are $7/3$-power-free have bounded nesting level, but this no longer holds for larger repetition exponents. We give an explicit characterization of the factors of the Thue-Mor...
Preprint
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The complement $\bar{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An antisquare is a nonempty word of the form $x\, \bar{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite bina...
Article
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For distinct vertices u and v in a graph G, the connectivity between u and v, denoted \(\kappa _G(u,v)\), is the maximum number of internally disjoint u–v paths in G. The average connectivity of G, denoted \({\overline{\kappa }}(G),\) is the average of \(\kappa _G(u,v)\) taken over all unordered pairs of distinct vertices u, v of G. Analogously, fo...
Preprint
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We construct an infinite binary word with critical exponent 3 that avoids abelian 4-powers. Our method gives an algorithm to determine if certain types of morphic sequences avoid additive powers. We also show that there are $\Omega(1.172^n)$ binary words of length $n$ that avoid abelian 4-powers, which improves on previous estimates.
Preprint
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Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$--$v$ paths in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined as $ \overline{\kappa}(G)=\sum_{\{u,v\}\subseteq V(G)} \kappa_G(u,v)/\tbinom{n}{2}. $ If $k \ge 1$ is an integer, then $...
Article
For a rational number r such that 1<r≤2, an undirected r-power is a word of the form xyx′, where the word x is nonempty, the word x′ is in {x,xR}, and we have |xyx′|/|xy|=r. The undirected repetition threshold for k letters, denoted URT(k), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonst...
Article
Let G be a (multi)graph of order n and let u,v be vertices of G. The maximum number of internally disjoint u–v paths in G is denoted by κG(u,v), and the maximum number of edge-disjoint u–v paths in G is denoted by λG(u,v). The average connectivity of G is defined by κ¯(G)=∑κG(u,v)∕n2, and the average edge-connectivity of G is defined by λ¯(G)=∑λG(u...
Article
An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every "extended" real...
Article
Let G be a graph, and let u, v, and w be vertices of G. If the distance between u and w does not equal the distance between v and w, then w is said to resolve u and v. The metric dimension of G, denoted β(G), is the cardinality of a smallest set W of vertices such that every pair of vertices of G is resolved by some vertex of W. The threshold dimen...
Article
The repetition threshold for words on n letters, denoted RT( n ), is the infimum of the set of all r such that there are arbitrarily long r -free words over n letters. A repetition threshold for circular words on n letters can be defined in three natural ways, which gives rise to the weak , intermediate , and strong circular repetition thresholds f...
Article
For a tree T, the subtree polynomial of T is the generating polynomial for the number of subtrees of T. We show that the complex roots of the subtree polynomial are contained in the disk z∈ℂ:|z|≤1+33, and that K1,3 is the only tree whose subtree polynomial has a root on the boundary. We also prove that the closure of the collection of all real root...
Article
For a graph G , the mean subtree order of G is the average order of a subtree of G . In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph G and every pair of distinct vertices u and v of G , the addition of the edge between u and v increases the mean subtree order. In...
Article
A variety of probabilistic notions of network reliability of graphs and digraphs have been proposed and studied since the early 1950s. Although grounded in the engineering and logistics of network design and analysis, the research also spans pure and applied mathematics, with connections to areas as diverse as combinatorics and graph theory, combin...
Preprint
Full-text available
An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real n...
Preprint
Full-text available
For a rational number $r$ such that $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where the word $x$ is nonempty, the word $x'$ is in $\{x,x^R\}$, and we have $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mbox{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are av...
Preprint
Full-text available
Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$, is the cardinality of a smallest set $W$ of vertices such that every pair of vertices of $G$ is resolved by s...
Preprint
Full-text available
A square-free word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word, and demonstrated that there are arbitrarily long extremal square-free ternary words. We find a...
Preprint
Full-text available
Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$, is the cardinality of a smallest set $W$ of vertices such that every pair of vertices of $G$ is resolved by s...
Preprint
Full-text available
The repetition threshold for words on $n$ letters, denoted $\mbox{RT}(n)$, is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetitio...
Preprint
Full-text available
For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every pair of distinct vertices $u$ and $v$ of $G$, the addition of the edge between $u$ and $v$ increases the mean sub...
Preprint
Full-text available
For every $n\geq 27$, we show that the number of $n/(n-1)^+$-free words (i.e., threshold words) of length $k$ on $n$ letters grows exponentially in $k$. This settles all but finitely many cases of a conjecture of Ochem.
Chapter
For rational \(1<r\le 2\), an undirected r-power is a word of the form \(xyx'\), where x is nonempty, \(x'\in \{x,{x}^{R}\}\), and \(|xyx'|/|xy|=r\). The undirected repetition threshold for k letters, denoted \(\mathrm {URT}(k)\), is the infimum of the set of all r such that undirected r-powers are avoidable on k letters. We first demonstrate that...
Preprint
Full-text available
A word of length $n$ is rich if it contains $n$ nonempty palindromic factors. An infinite word is rich if all of its finite factors are rich. Baranwal and Shallit produced an infinite binary rich word with critical exponent $2+\sqrt{2}/2$ and conjectured that this was the least possible critical exponent for infinite binary rich words (i.e., that t...
Preprint
Full-text available
For distinct vertices $u$ and $v$ in a graph $G$, the {\em connectivity} between $u$ and $v$, denoted $\kappa_G(u,v)$, is the maximum number of internally disjoint $u$--$v$ paths in $G$. The {\em average connectivity} of $G$, denoted $\overline{\kappa}(G),$ is the average of $\kappa_G(u,v)$ taken over all unordered pairs of distinct vertices $u,v$...
Article
Full-text available
A word is called $\beta$-free if it has no factors of exponent greater than or equal to $\beta$. The repetition threshold $\mbox{RT}(k)$ is the infimum of the set of all $\beta$ such that there are arbitrarily long $k$-ary $\beta$-free words (or equivalently, there are $k$-ary $\beta$-free words of every sufficiently large length, or even every len...
Preprint
Full-text available
For rational $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where $x\neq \varepsilon$, $x'\in\{x,x^R\}$, and $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mathrm{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that...
Article
This article focuses on properties and structures of trees with maximum mean subtree order in a given family; such trees are called optimal in the family. Our main goal is to describe the structure of optimal trees in and , the families of all trees and caterpillars, respectively, of order . We begin by establishing a powerful tool called the Gluin...
Preprint
Full-text available
The average order of the connected induced subgraphs of a graph $G$ is called the mean connected induced subgraph (CIS) order of $G$. This is an extension of the mean subtree order of a tree, first studied by Jamison. In this article, we demonstrate that among all connected block graphs of order $n$, the path $P_n$ has minimum mean CIS order. This...
Preprint
Full-text available
For a tree $T$, the subtree polynomial of $T$ is the generating polynomial for the number of subtrees of $T$. We show that the complex roots of the subtree polynomial are contained in the disk $\left\{z\in\mathbb{C}\colon\ |z|\leq 1+\sqrt[3]{3}\right\}$, and that $K_{1,3}$ is the only tree whose subtree polynomial has a root on the boundary. We als...
Preprint
Full-text available
Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted by $\lambda_G (u,v)$. The average connectivity of $G$ is defined by $\overline{\kappa}(G)=\sum_{\{u,v\}\subsete...
Preprint
Full-text available
We make certain bounds in Krebs' proof of Cobham's theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic $a$-automatic sequence and an aperiodic $b$-automatic sequence, where $a$ and $b$ are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors i...
Article
Full-text available
A word is called $\beta$-free if it has no factors of exponent greater than or equal to $\beta$. The repetition threshold $RT(k)$ is the infimum of the set of all $\beta$ such that there are arbitrarily long $k$-ary $\beta$-free words (or equivalently, there are $k$-ary $\beta$-free words of every sufficiently large length, or even every length). T...
Article
Full-text available
The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the sum is taken over all unordered pairs of distinct vertices of $G$. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the ma...
Article
Full-text available
In the interest of studying formulas with reversal of high avoidability index, we find $n$-avoidance bases for formulas with reversal for $n\in\{1,2,3\}$. We demonstrate that there is a unique formula with reversal in each of these three bases of highest avoidability index $n+2$; these formulas are $xx$, $xyx\cdot y^R$, and $xyzx\cdot y^R\cdot z^R$...
Article
Full-text available
In this article the extremal structures for the mean order of connected induced subgraphs of cographs are determined. It is shown that among all connected cographs of order $n \ge 7$, the star $K_{1,n-1}$ has maximum mean connected induced subgraph order, and for $n \ge 3$, the $n$-skillet, $K_1+(K_1 \cup K_{n-2})$, has minimum mean connected induc...
Article
Full-text available
This article focuses on properties and structures of trees with maximum mean subtree order in a given family; such trees are called optimal in the family. For a fixed tree $Q$ with root vertex $v$, it is shown that among all trees obtained by gluing $v$ to a vertex of a path, the optimal tree occurs if $v$ is glued to a central vertex of the path....
Article
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Given a graph $G$ in which each edge fails independently with probability $q\in[0,1],$ the all-terminal reliability of $G$ is the probability that all vertices of $G$ can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in $q$ whose roots (all-terminal rel...
Article
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While a characterization of unavoidable formulas (without reversal) is well-known, little is known about the avoidability of formulas with reversal in general. In this article, we characterize the unavoidable formulas with reversal that have at most two one-way variables ($x$ is a one-way variable in formula with reversal $\phi$ if exactly one of $...
Article
Full-text available
Given a graph $G$ whose edges are perfectly reliable and whose nodes each operate independently with probability $p\in[0,1],$ the node reliability of $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce. We study analytic properties of the node reliability o...
Article
Full-text available
We present an infinite family of formulas with reversal whose avoidability index is bounded between (Formula presented.) and (Formula presented.) and we show that several members of the family have avoidability index (Formula presented.) This family is particularly interesting due to its size and the simple structure of its members. For each (Formu...
Article
Full-text available
Given a graph $G$ whose edges are perfectly reliable and whose nodes each operate independently with probability $p\in[0,1],$ the node reliability of $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce; it is the analogous node measure of robustness to the...

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Project (1)
Project
This project is concerned with the study of problems related to Dejean's conjecture (now Dejean's theorem).