arXiv subjects

Aida Abiad

Publications and source records attributed to Aida Abiad.

At least 37 records · Page 2Linked to original sources

arXiv2026

Eigenvalue bounds for distance-edge colorings

For a fixed positive integer $t$, we consider the graph colouring problem in which edges at distance at most $t$ are given distinct colours. We obtain sharp lower bounds for the distance-$t$ chromatic index, the least number of colours necessary for such a colouring. Our bounds are of algebraic nature; they depend on the eigenvalues of the line graph and on a polynomial which can be found using integer linear programming methods. We show several graph classes that attain equality for our bounds, and also present some computational results which illustrate the bound's performance. Lastly, we investigate the implications the spectral approach has to the Erdős-Nešetřil conjecture, and derive some conditions which a graph must satisfy if we could use it to obtain a counter example through the proposed spectral methods.

math.CO↗

arXiv2026

On the Diameter of Arrangements of Topological Disks

Let $\mathcal{D}=\{D_0,\ldots,D_{n-1}\}$ be a set of $n$ topological disks in the plane and let $\mathcal{A} := \mathcal{A}(\mathcal{D})$ be the arrangement induced by $\mathcal{D}$. For two disks $D_i,D_j\in\mathcal{D}$, let $Δ_{ij}$ be the number of connected components of $D_i\cap D_j$, and let $Δ:= \max_{i,j} Δ_{ij}$. We show that the diameter of $\mathcal{G}^*$, the dual graph of $\mathcal{A}$, can be bounded as a function of $n$ and $Δ$. Thus, any two points in the plane can be connected by a Jordan curve that crosses the disk boundaries a number of times bounded by a function of $n$ and $Δ$. In particular, for the case of two disks, we prove that the diameter of $\mathcal{G}^*$ is at most $\max\{2,2Δ\}$ and this bound is tight. For the general case of $n>2$ disks, we show that the diameter of $\mathcal{G}^*$ is $O(n^3 2^n Δ)$. We achieve this by proving that the number of maximal faces in $\mathcal{A}$ -- faces whose ply is more than the ply of their neighboring faces -- is $O(n^2 2^n Δ)$. To this end, we first show that the number of maximum faces -- faces whose ply is $n$ -- is $O(n^2Δ)$; the latter bound, which is of independent interest, is tight in the worst case.

math.CO↗

arXiv2026

Logical aspects of isomorphism of controllable graphs and cospectrality of distance-regularized graphs

We consider isomorphism of controllable graphs and cospectrality of distance-regularized graphs (which are known to be distance-regular or distance-biregular) in relation to logical definability. While most characterizations of these equivalence relations for such graph classes are of algebraic and spectral flavor, here we inject tools from first-order logic, extending and unifying several existing results.

math.CO↗

arXiv2026

A strong nullity parameter for rooted graphs

The inverse eigenvalue problem of a graph $G$ studies the possible spectra of matrices associated with $G$, including as an important subproblem the possible nullities of such a matrix. Much research in this area to date has focused only on the spectrum of the matrix itself, but there are applications of inverse eigenvalue problems that also involve the interaction between that spectrum and the spectrum of some maximal proper principal submatrix, or in other words the interlacing spectrum that results from crossing out any one row and the same column. Motivated by this refined information, given a graph $G$ on $n$ vertices with a designated root vertex, we investigate all possible nullity pairs where the first nullity is that of an $n \times n$ symmetric matrix associated to $G$ and the second nullity is that of the principal submatrix of size $(n - 1) \times (n - 1)$ that results from deleting the row and column associated to the root vertex. We define a new parameter $ξξ(G,i)$ for rooted graphs $(G,i)$ equipped with the strong nullity interlacing property that coordinates the two values of a nullity pair, and we show that this graph parameter is minor monotone. Moreover, we prove a bifurcation lemma for the strong nullity interlacing property. We use these new tools to characterize the rooted graphs with $ξξ(G,i) \geq s$ for $s \in \{ 0,1,2,3,4, 5\}$ by finding the minimal minors for each of these families. These families turn out to have strong connections to the minimal minors for $ξ(G) \geq k$.

math.CO↗

arXiv2026

Propagation processes on (hyper)graphs: where zero forcing and burning meet

The burning and forcing processes are both instances of propagation processes on graphs that are commonly used to model real-world spreading phenomena. The contribution of this paper is two-fold. We first establish a connection between these two propagation processes via hypergraphs. We do so by showing a sharp upper bound on the zero forcing number of the incidence graph of a hypergraph in terms of the lazy burning number of the hypergraph, which builds up on and improves a result by Bonato, Jones, Marbach, Mishura and Zhang (Theor. Comput. Sci., 2025). Secondly, we deepen the understanding of the role of the burning process in the context of graph spectral characterizations, whose goal is to understand which graph properties are encoded in the spectrum. While for several graph properties, including the zero forcing number, it is known that the spectrum does not encode them, this question remained open for the burning number. We solve this problem by constructing infinitely many pairs of cospectral graphs which have a different burning number.

math.CO↗

arXiv2025

Improved Gilbert-Varshamov bound for sum-rank-metric codes via graph theory

We use a graph-theoretic approach which yields improvements on the known Gilbert-Varshamov (GV) bound for sum-rank-metric codes for certain parameters. In particular, we show that asymptotically $\mathbb{F}_q^{\mathbf{n} \times \mathbf{m}}$ can be partitioned into sum-rank-metric codes whose average size is bigger than the GV bound by a logarithmic factor for these parameters. Finally, we discuss the connection of such codes to set-coloring Ramsey numbers.

math.CO↗

arXiv2025

Tales of Hoffman: from a distance

Hoffman proved that a graph $G$ with adjacency eigenvalues $λ_1\geq \cdots \geq λ_n$ and chromatic number $χ(G)$ satisfies $χ(G)\geq 1+κ,$ where $κ$ is the smallest integer such that $$λ_1+\sum_{i=1}^κλ_{n+1-i}\leq 0.$$ We extend this eigenvalue bound to the distance-$k$ setting, and also show a strengthening of it by proving that it also lower bounds the corresponding quantum distance coloring graph parameter. The new bound depends on a degree-$k$ polynomial which can be chosen freely, so one needs to make a good choice of the polynomial to obtain as strong a bound as possible. We thus propose linear programming methods to optimize it. We also investigate the implications of the new bound for the quantum distance chromatic number, showing that it is sharp for some classes of graphs. Finally, we extend the Hoffman bound to the distance setting of the vector chromatic number. Our results extend and unify several previous bounds in the literature.

math.CO↗

arXiv2025

The Shannon capacity of graph powers

For a graph $G$, its $k$-th graph power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$. We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral graph theory and linear optimization methods. First, we use the so-called ratio-type bound to provide an alternative and spectral proof of a result by Lovász [IEEE Trans. Inform. Theory 1979], which states that, for a regular graph, the Hoffman ratio bound on the independence number is also an upper bound on the Lovász theta number and, hence, also on the Shannon capacity. In fact, we show that Lovász' result holds in the more general context of graph powers. Secondly, we derive another bound on the Shannon capacity of graph powers, the so-called rank-type bound, which depends on a new family of polynomials that can be computed by running a simple algorithm. Lastly, we provide several computational experiments that demonstrate the sharpness of the two proposed algebraic bounds. As a byproduct, when these two new algebraic bounds are tight, they can be used to easily derive the exact values of the Lovász theta number (which relies on solving an SDP) and the Shannon capacity (which is not known to be computable) of the corresponding graph power.

math.CO↗

arXiv2025

Counting cospectral graphs obtained via switching

Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their spectrum. Hence, switching provides a tool for disproving the existence of such a characterization. This paper presents a general framework for counting the number of graphs that have a non-isomorphic cospectral graph through a switching method, expanding on the work by Haemers and Spence [European Journal of Combinatorics, 2004]. Our framework is based on a different counting approach, which allows it to be used for all known switching methods for the adjacency matrix. From this, we derive asymptotic results, which we complement with computer enumeration results for graphs up to $10$ vertices.

math.CO↗

arXiv2025

The Eigenvalue Method in Coding Theory

We lay down the foundations of the Eigenvalue Method in coding theory. The method uses modern algebraic graph theory to derive upper bounds on the size of error-correcting codes for various metrics, addressing major open questions in the field. We identify the core assumptions that allow applying the Eigenvalue Method, test it for multiple well-known classes of error-correcting codes, and compare the results with the best bounds currently available. By applying the Eigenvalue Method, we obtain new bounds on the size of error-correcting codes that often improve the state of the art. Our results show that spectral graph theory techniques capture structural properties of error-correcting codes that are missed by classical coding theory approaches.

math.CO↗

arXiv2025

A graph energy conjecture through the lenses of semidefinite programming

Let $G$ be a graph on $n$ vertices with independence number $α(G)$. Let $\mathcal{E}(G)$ be the energy of a graph, defined as the sum of the absolute values of the adjacency eigenvalues of $G$. Using Graffiti, Fajtlowicz conjectured in the 1980s that $$\frac{1}{2}\mathcal{E}(G) \geq n - α(G).$$ In this paper we derive a semidefinite program formulation of the graph energy, and we use it to obtain several results that constitute a first step towards proving this conjecture. In particular, we show that $$\frac{1}{2}\mathcal{E}(G) \geq n - χ_f(\overline{G}) \quad \text{ and } \quad \frac{1}{2}\mathcal{E}(G) \geq n - H(G),$$ where $χ_f(G)$ is the fractional chromatic number and $H(G)$ is Hoffman's ratio number. As a byproduct of the SDP formulation we obtain several lower bounds for the graph energy that improve and refine previous results by Hoffman (1970) and Nikiforov (2007). The later author showed that the conjecture holds for almost all graphs. However, the graph families known to attain the conjecture with equality are highly structured and do not represent typical graphs. Motivated by this, we prove the following bound in support of the conjecture for the class of highly regular graphs $$\frac{1}{2}\mathcal{E}(G) \geq n - \vartheta^-(G),$$ where $\vartheta^-$ is Schrijver's theta number.

math.CO↗

arXiv2025

On the $Z_q$-forcing number: computational approach and exact values

Zero forcing is a graph coloring process that is used to model spreading phenomena in real-world scenarios. It can also be viewed as a single-player combinatorial game on a graph, where the player's goal is to select a subset of vertices of minimum cardinality that eventually leads to all vertices of the graph being colored. A variant of this game, called the $q$-analogue of zero forcing, was later introduced. In this version, the player again seeks to choose the smallest number of vertices that will eventually color the entire graph, while an oracle attempts to force the player to select a larger subset. In this paper, we exploit the structural properties of several graph classes in order to both derive algorithms to compute the exact value of $Z_q$, and to establish bounds and exact values of the parameter for these graph classes. In particular, we present a SAT-based algorithm to compute the $q$-analogue zero forcing number, offering optimal strategies for both the player and the oracle. Additionally, we propose a polynomial-time algorithm for computing the $q$-analogue zero forcing number for $q=1$ of cactus graphs. Lastly, we prove the exact value of this parameter for several graph classes such as block graphs. Our work extends previous results about trees by Butler et al. (2020) and Blanco et al. (2024).

math.CO↗

arXiv2025

Hoffman colorability of (strongly) regular graphs

Hoffman's bound is a well-known eigenvalue bound on the chromatic number of a graph. By interpreting this bound as a parameter, we show multiple applications of colorings attaining the bound (Hoffman colorings) for several notions of graph regularity: regular, (co-)edge-regular, and strongly regular. For strongly regular graphs, we prove that Hoffman colorability implies pseudo-geometricity, and we strengthen Haemers' finiteness result on strongly regular graphs with a bounded chromatic number by considering the Hoffman bound instead of the chromatic number. Furthermore, by using Hoffman colorings we show that a sufficient condition for non-unique vector colorability shown by Godsil, Roberson, Rooney, Šámal and Varvitsiotis [European J. Combin. 79, 2019] can be relaxed in the setting of strongly regular graphs. Lastly, using Hoffman colorings we derive several new characterizations of the mentioned graph regularity notions.

math.CO↗

arXiv2025

On the sum of the largest and smallest eigenvalues of odd-cycle free graphs

Let $G$ be a graph with adjacency eigenvalues $λ_1 \geq \cdots \geq λ_n$. Both $λ_1 + λ_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikvári proved in 2022 that for odd girth 5 graphs (triangle-free) it holds that $(λ_1+λ_n)/n \le (3-2\sqrt 2) < 0.1716$. In this paper we extend Csikvári's result to general odd girth $k$ proving that $(λ_1+λ_n)/n = O(k^{-1})$. In the case of odd girth 7, we prove a stronger upper bound of $(λ_1+λ_n)/n < 0.0396$.

math.CO↗

arXiv2025

Infinitely many counterexamples to a conjecture of Lovász

Motivated by the well-known conjecture of Ryser which relates maximum matchings to minimum vertex covers in $r$-partite $r$-uniform hypergraphs, Lovász formulated a stronger conjecture. It states that one can always reduce the matching number by removing $r-1$ vertices. This conjecture was very recently disproven for $r=3$ by Clow, Haxell, and Mohar using the line graph of a $3$-regular graph of order $102$. Building on this, we describe a simple infinite family of counterexamples based on generalized Petersen graphs for the case $r=3$ and give specific counterexamples for $r=4$.

math.CO↗

arXiv2025

A Spectral Approach to Kemeny's Constant

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, providing insight into the global behavior of a Markov chain. We present a novel eigenvector-based formula for computing Kemeny's constant. Moreover, we analyze the impact of network structure on Kemeny's constant. In particular, we use various spectral techniques, such as spectral sparsification of graphs and eigenvalue interlacing, and show that they are particularly useful in this context for deriving approximations and sharp bounds for Kemeny's constant

math.CO↗

arXiv2025

Eigenvalue bounds for the quantum chromatic number of graph powers

The quantum chromatic number, a generalization of the chromatic number, was first defined in relation to the non-local quantum coloring game. We generalize the former by defining the quantum $k$-distance chromatic number $χ_{kq}(G)$ of a graph $G$, which can be seen as the quantum chromatic number of the $k$-th power graph, $G^k$, and as generalization of the classical $k$-distance chromatic number $χ_k(G)$ of a graph. It can easily be shown that $χ_{kq}(G) \leq χ_k(G)$. In this paper, we strengthen three classical eigenvalue bounds for the $k$-distance chromatic number by showing they also hold for the quantum counterpart of this parameter. This shows that several bounds by Elphick et al. [J. Combinatorial Theory Ser. A 168, 2019, Electron. J. Comb. 27(4), 2020] hold in the more general setting of distance-$k$ colorings. As a consequence we obtain several graph classes for which $χ_{kq}(G)=χ_{k}(G)$, thus increasing the number of graphs for which the quantum parameter is known.

math.CO↗

arXiv2025

Hoffman colorings of graphs

Hoffman's bound is a well-known spectral bound on the chromatic number of a graph, known to be tight for instance for bipartite graphs. While Hoffman colorings (colorings attaining the bound) were studied before for regular graphs, for general graphs not much is known. We investigate tightness of the Hoffman bound, with a particular focus on irregular graphs, obtaining several results on the graph structure of Hoffman colorings. In particular, we prove a Decomposition Theorem, which characterizes the structure of Hoffman colorings, and we use it to completely classify Hoffman colorability of cone graphs and line graphs. We also prove a partial converse, the Composition Theorem, leading to an algorithm for computing all connected Hoffman colorable graphs for some given number of vertices and colors. Since several graph coloring parameters are known to be sandwiched between the Hoffman bound and the chromatic number, as a byproduct of our results, we obtain the values of these chromatic parameters.

math.CO↗