An eigenvalue of a graph is called main if its eigenspace is not orthogonal to the all-ones vector. Introduced by Cvetkovi\'{c} in the early 1970s and systematically studied by Rowlinson and others, graphs with exactly one or two main eigenvalues are now well understood. However, the classification of graphs with precisely three main eigenvalues remains a challenging open problem in spectral graph theory. This paper provides a complete classification of all trees of diameter 5 with exactly three main eigenvalues. Using equitable partitions, the spectral condition reduces to the unique solvability of linear systems over the rationals, leading to Diophantine equations involving branch lengths and pendant counts. We prove that every such tree is isomorphic either to a symmetric tree $T_r(a)$ or to a member of a parametric family $\mathcal{T}$ determined by arithmetic divisibility conditions. We also construct an infinite family of such trees with unbounded diameter.
A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\cos(\pi/9)]$ and one in $[-2\cos(\pi/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio...
R. O. Braga, Jean Carlo Moraes, Matheus C. Santos· 0 citations
An eigenvalue of a signed graph is called \emph{main} if there exists a corresponding eigenvector non-orthogonal to the all-ones vector. An important result of O'Rourke and Touri (2016) states that almost all (unsigned) graphs have all main eigenvalues. Akbari, Fran\c{c}a, Ghasemian, Javarsineh, and de Lima (2021) cons...
S. Akbari, Hitesh Kumar, Bojan Mohar et al.· 0 citations
We study the negative spectrum of the Laplacian on a metric graph with general vertex matching conditions and with two length scales: a compact core whose edges have length of order a small parameter $\epsilon$, together with finitely many edges of infinite length. As $\epsilon\to0$, some negative eigenvalues may escap...
Gregory Berkolaiko, Denis I. Borisov, Marshall King et al.· 0 citations
For a graph \(G\) admitting a real symmetric realization with exactly two distinct eigenvalues, \(MB(G)\) is the minimum, over all such realizations, of the smaller of the two eigenvalue multiplicities. Adm, Fallat, Meagher, Nasserasr, Plosker, and Yang asked for this parameter for the complement of a path on at least...
In this paper, we investigate the eigenvalues of character degree graphs, with particular emphasis on the arithmetic properties of their spectra. First, we study \((n-2)\)-regular character degree graphs of solvable groups and derive an explicit formula for their characteristic polynomials. We show that all their eigen...
G. Sivanesan, C. Selvaraj, J. Laubacher· 0 citations
In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eige...
Benedikt Kolbe, Jack Spalding-Jamieson· 0 citations
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