arXiv subjects

Luis M. B. Varona

Publications and source records attributed to Luis M. B. Varona.

4 recordsLinked to original sources

arXiv2026

Enumeration of Laplacian integral and {-1,0,1}-diagonalizable graphs

A graph with Laplacian matrix $L$ is called Laplacian integral if the eigenvalues of $L$ are all integers, and it is called $\{-1,0,1\}$-diagonalizable if $L$ has a full set of eigenvectors with entries from $\{-1,0,1\}$. We herein develop a structure theorem for both Laplacian integral graphs and $\{-1,0,1\}$-diagonalizable graphs of prime order, and combine it with some novel computational techniques to characterize all such graphs for orders larger than was previously possible. For example, we enumerate all Laplacian integral and $\{-1,0,1\}$-diagonalizable graphs of order $13$ or less, all $\{-1,0,1\}$-diagonalizable graphs of prime order $23$ or less, all regular integral graphs of order $15$ or less, and all regular $\{-1,0,1\}$-diagonalizable graphs of prime order $53$ or less. As an immediate byproduct of our work, we show that the $S_{n,n}$ conjecture for Laplacian integral graphs is true when $n = 12$, thus making $n = 16$ the smallest open case; additionally, we disprove two related conjectures regarding Laplacian spectra. We also establish an exponential lower bound on the number of connected $\{-1,0,1\}$-diagonalizable graphs of order $n$, thus beating the previously best-known (subexponential) lower bound. Finally, we show that every bipartite $\{-1,0,1\}$-diagonalizable graph is regular (a fact that fails to generalize to Laplacian integral graphs).

math.CO↗

arXiv2026

A polynomial-time algorithm for recognizing high-bandwidth graphs

An unweighted, undirected graph $G$ on $n$ nodes is said to have \emph{bandwidth} at most $k$ if its nodes can be labelled from $0$ to $n - 1$ such that no two adjacent nodes have labels that differ by more than $k$. It is known that one can decide whether the bandwidth of $G$ is at most $k$ in $O(n^k)$ time and $O(n^k)$ space using dynamic programming techniques. For small $k$ close to $0$, this approach is effectively polynomial, but as $k$ scales with $n$, it becomes superexponential, requiring up to $O(n^{n - 1})$ time (where $n - 1$ is the maximum possible bandwidth). In this paper, we reformulate the problem in terms of bipartite matching for sufficiently large $k \ge \lfloor (n - 1)/2 \rfloor$, allowing us to use Hall's marriage theorem to develop an algorithm that runs in $O(n^{n - k + 1})$ time and $O(n)$ auxiliary space (beyond storage of the input graph). This yields polynomial complexity for large $k$ close to $n - 1$, demonstrating that the bandwidth recognition problem is solvable in polynomial time whenever either $k$ or $n - k$ remains small.

cs.DS↗

arXiv2026

Efficient spectral bounds on the chromatic number of Hamming, Johnson, and Kneser graph powers

We investigate spectral lower bounds on the chromatic number $χ$ of Hamming graph powers $H(n, q)^p$, Johnson graph powers $J(n, k)^p$, and Kneser graph powers $K(n, k)^p$ providing the first computationally feasible nontrivial results. While the classical Hoffman bound on $χ$ can, in principle, be applied to any graph, naïve computation requires $O(q^{3n})$ time for $H(n, q)^p$ and $O(({}_nC_k)^3)$ time for both $J(n, k)^p$ and $K(n, k)^p$. We thus express the adjacency eigenvalues of these graphs in terms of hypergeometric orthogonal polynomials, exploiting recurrence relations that arise to efficiently compute the entire spectra. We then apply dynamic programming to compute the Hoffman bounds for $H(n, q)^p$, $J(n, k)^p$, and $K(n, k)^p$ in $O(np)$, $O(kp)$, and $O(k^2)$ time, respectively.

math.CO↗

arXiv2025

Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\|\mathbf{v}^2\|\|\mathbf{w}^2\| - \langle\mathbf{v}^2,\mathbf{w}^2\rangle \leq \|\mathbf{v}\|^2\|\mathbf{w}\|^2 - \langle\mathbf{v},\mathbf{w}\rangle^2$ for all $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$. We present three new proofs of this inequality that better illustrate "why" it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we derive a version of the inequality involving three or more vectors.

math.FA↗