Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.19981

Co-edge-regular graphs with four eigenvalues and unbounded coherent rank

Edwin R. van Dam·Hong-Jun Ge·Jack H. Koolen
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the regular three-eigenvalue setting, spectral complexity and coherent-algebraic complexity coincide: a connected regular graph has exactly three distinct eigenvalues if and only if it is strongly regular, its coherent rank is three. Although examples of regular graphs with four distinct eigenvalues and coherent rank larger than four are known, it was unknown whether coherent rank is uniformly bounded among regular graphs with four distinct eigenvalues. We show that no such bound exists, even under the additional assumption of co-edge-regularity. For every prime power \(q\), we construct infinitely many co-edge-regular graphs with exactly four distinct eigenvalues, smallest eigenvalue \(-2q-1\), and coherent rank at least \(q+4\). Consequently, coherent rank is unbounded among co-edge-regular graphs with exactly four distinct eigenvalues.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Edwin R. van Dam, Hong-Jun Ge, Jack H. Koolen. 2026-06-18. Co-edge-regular graphs with four eigenvalues and unbounded coherent rank. https://arxiv.org/abs/2606.19981

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗