Author ORCID Identifier
0009-0006-2154-3372
Date of Award
8-2024
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics and Statistics
First Advisor
Guantao Chen
Second Advisor
Yaroslav Molkov
Abstract
In this dissertation, three major topics in graph theory and computational neuroscience are explored: linear arboricity of graphs, $ f $-density parameter in fractional graph edge coloring, and quadruped locomotion gait with sensory feedback and central interactions. A linear forest is a disjoint union of path graphs. The \textit{linear arboricity} of a simple graph $ G $, denoted by $ \operatorname{la}(G) $, is the least number of linear forests into which the graph can be partitioned. The long-standing \textit{Linear Arboricity Conjecture} (LAC) from 1981 asserts that $ \operatorname{la}(G) \le \lceil (\Delta(G)+1)/2 \rceil $. In Chapter~\ref{chap:LAC}, two methods are introduced and the LAC is proved for several different graph classes, especially degenerate graphs with relatively large maximum degree. Let $f$ be a function from $V(G)$ to $\mathbb{Z}_+$. The {\em fractional $f$-density} $\mathcal{W}^*_{f}$ of a loopless multigraph $G$ is defined as: $$ \mathcal{W}^*_{f}(G)=\max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor f(U) / 2\rfloor}, $$ where $ f(U)=\sum_{v\in U} f(v) $. It is a generalization of the density parameter in graph edge coloring problem which plays an important role in determining the $ f $-chromatic index. In Chapter~\ref{chap:density}, a polynomial-time algorithm is given for calculating $\mathcal{W}^*_{f}(G)$ in terms of the number of vertices of $G$. \textit{Quadrupeds locomotion} is a complex process involving specific interactions between the central neural controller and the mechanical components of the system. In Chapter~\ref{chap:locomotion}, a tractable mathematical model of mouse locomotion are created and analyzed using biomechanical data and recent findings on the organization of neural interactions within the spinal locomotor circuitry. Several model versions are investigated and compared to existing experimental data on mouse locomotion. The results highlight the specific roles of sensory feedback and some central propriospinal interactions between circuits controlling fore and hind limbs for speed-dependent gait expression. The models suggest that postural imbalance feedback may be critically involved in the control of swing-to-stance transitions in each limb and the stabilization of walking direction.
DOI
https://doi.org/10.57709/37140947
Recommended Citation
Yu, Guoning, "Topics in Graph Theory and Computational Neuroscience — Linear Arboricity, Graph f-density and Quadruped Locomotion." Dissertation, Georgia State University, 2024.
doi: https://doi.org/10.57709/37140947
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