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- W4377130536 abstract "For an $n times n$ matrix $A$, let $q(A)$ be the number of distinct eigenvalues of $A$. If $G$ is a connected graph on $n$ vertices, let $mathcal{S}(G)$ be the set of all real symmetric $n times n$ matrices $A=[a_{ij}]$ such that for $ineq j$, $a_{ij}=0$ if and only if ${i,j}$ is not an edge of $G$. Let $q(G)={rm min}{q(A),:,A in mathcal{S}(G)}$. Studying $q(G)$ has become a fundamental sub-problem of the inverse eigenvalue problem for graphs, and characterizing the case for which $q(G)=2$ has been especially difficult. This paper considers the problem of determining the regular graphs $G$ that satisfy $q(G)=2$. The resolution is straightforward if the degree of regularity is $1, 2,$ or $3$. However, the $4$-regular graphs with $q(G)=2$ are much more difficult to characterize. A connected $4$-regular graph has $q(G)=2$ if and only if either $G$ belongs to a specific infinite class of graphs, or else $G$ is one of fifteen $4$-regular graphs whose number of vertices ranges from $5$ to $16$. This technical result gives rise to several intriguing questions." @default.
- W4377130536 created "2023-05-21" @default.
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- W4377130536 date "2023-05-17" @default.
- W4377130536 modified "2023-09-26" @default.
- W4377130536 title "Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues" @default.
- W4377130536 doi "https://doi.org/10.48550/arxiv.2305.10562" @default.
- W4377130536 hasPublicationYear "2023" @default.
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