arXiv subjects

Johnna Parenteau

Publications and source records attributed to Johnna Parenteau.

3 recordsLinked to original sources

arXiv2026

Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach

For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we provide an alternative technique to establish that $q(G) \leq 4$ for any threshold graph $G$ as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further

math.CO↗

arXiv2026

Strictly Interlaced Spectral Data for the Weighted Matching Polynomial of a Graph

Interlacing of the real roots of a weighted matching polynomial for a graph $G$ and that of a vertex-deleted subgraph is classical and well-known. In the context of strict interlacing of distinct roots, a demonstrated graph construction gives rise to a new classification of graphs, called ${ \rm SRSI}$ graphs, which include graphs that contain a Hamilton path. Graphs with a perfect (or nearly perfect) matching are shown to exhibit the SRSI$_w(v)$ property with respect to a particular weighting and for specific vertices, and all such graphs are characterized via this graph construction. As a consequence, we also characterize the trees that possess the SRSI property for all edge weightings and all vertices.

math.CO↗

arXiv2024

Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs

We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every Johnson graph has a signed variant with exactly two distinct eigenvalues. We also explore applications to weighing matrices, linear ternary codes, tight frames, and compute the minimum rank of Johnson graphs. Further results involve the minimum number of distinct eigenvalues for graphs in association schemes, distance-regular graphs, and Hamming graphs. We also draw some connections with simplicial complexes and higher-order Laplacians.

math.CO↗