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Guangjun Xu

Publications and source records attributed to Guangjun Xu.

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Democratizing Music Therapy: LLM-Based Automated EEG Analysis and Progress Tracking for Low-Cost Home Devices

Home-based music therapy devices require accessible and cost-effective solutions for users to understand and track their therapeutic progress. Traditional physiological signal analysis, particularly EEG interpretation, relies heavily on domain experts, creating barriers to scalability and home adoption. Meanwhile, few experts are capable of interpreting physiological signal data while also making targeted music recommendations. While large language models (LLMs) have shown promise in various domains, their application to automated physiological report generation for music therapy represents an unexplored task. We present a prototype system that leverages LLMs to bridge this gap -- transforming raw EEG and cardiovascular data into human-readable therapeutic reports and personalized music recommendations. Unlike prior work focusing on real-time physiological adaptation during listening, our approach emphasizes post-session analysis and interpretable reporting, enabling non-expert users to comprehend their psychophysiological states and track therapeutic outcomes over time. By integrating signal processing modules with LLM-based reasoning agents, the system provides a practical and low-cost solution for short-term progress monitoring in home music therapy contexts. This work demonstrates the feasibility of applying LLMs to a novel task -- democratizing access to physiology-driven music therapy through automated, interpretable reporting.

cs.HC

Hadwiger's conjecture for the complements of Kneser graphs

Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such that two vertices are adjacent if and only if the corresponding $k$-subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.

math.CO

Three-arc graphs: characterization and domination

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we give a characterization of 3-arc graphs and obtain sharp upper bounds on the domination number of the 3-arc graph of a graph $G$ in terms that of $G$.

math.CO

Symmetric graphs with 2-arc transitive quotients

A graph $\Ga$ is $G$-symmetric if $\Ga$ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Ga$, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\Ga)$, namely when $V(\Ga)$ admits a nontrivial $G$-invariant partition $\BB$, the quotient graph $\Ga_{\BB}$ of $\Ga$ with respect to $\BB$ is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for $\Ga_{\BB}$ to be $(G, 2)$-arc transitive (regardless of whether $\Ga$ is $(G, 2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of $\BB$ and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v, k$ and two other parameters with respect to $(\Ga, \BB)$ together with a certain 2-point transitive block design induced by $(\Ga, \BB)$. We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for $\Ga_{\BB}$ to be $(G, 2)$-arc transitive.

math.CO

Hamiltonicity of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are Hamiltonian. As a consequence we obtain that if a vertex-transitive graph is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three, then it is Hamiltonian. This confirms the well known conjecture, that all vertex-transitive graphs with finitely many exceptions are Hamiltonian, for a large family of vertex-transitive graphs. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.

math.CO

Hadwiger's conjecture for 3-arc graphs

The 3-arc graph of a digraph $D$ is defined to have vertices the arcs of $D$ such that two arcs $uv, xy$ are adjacent if and only if $uv$ and $xy$ are distinct arcs of $D$ with $v\ne x$, $y\ne u$ and $u,x$ adjacent. We prove that Hadwiger's conjecture holds for 3-arc graphs.

math.CO