## 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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