arXiv subjects

Raffaella Mulas

Publications and source records attributed to Raffaella Mulas.

At least 19 recordsLinked to original sources

arXiv2026

Cheeger-type inequalities for the second largest spectral gap from $1$ of the normalized Laplacian

We study the second largest spectral gap from $1$ of the normalized Laplacian of a graph, a quantity that appears in the literature in connection with random walks, expander graphs, and Ramanujan graphs. We relate it to the classical Cheeger and dual Cheeger constants, and we introduce a new Cheeger-type constant admitting a probabilistic interpretation in terms of two-step random walks. For this constant, we establish sharp inequalities analogous to the classical Cheeger inequalities.

math.CO↗

arXiv2026

Spectral Perspectives on FAT Graph Colorings

We investigate Fair and Tolerant (FAT) graph colorings, a coloring framework in which each vertex is allowed to share its color with a prescribed fraction of its neighbors, while the remaining neighbors are required to be distributed evenly among the other coloring classes. In particular, we determine the FAT chromatic number for all complete multipartite graphs, and we analyze the behavior of FAT colorings under several graph operations. Although spectral methods form the primary focus, several combinatorial arguments are included to complement the results.

math.CO↗

arXiv2025

Fair and Tolerant (FAT) Graph Colorings

We introduce and study Fair and Tolerant colorings (FAT colorings), where each vertex tolerates a given fraction of same-colored neighbors while fairness is preserved across the other coloring classes. Moreover, we define the FAT chromatic number $\chi^{\mathrm{FAT}}(G)$ as the largest integer $k$ for which $G$ admits a FAT $k$-coloring. We establish general bounds on $\chi^{\mathrm{FAT}}$, relate it to structural and spectral properties of graphs, and characterize it completely for several families of graphs. We conclude with a list of open questions that suggest future directions.

math.CO↗

arXiv2025

Coloring outside the lines: Spectral bounds for generalized hypergraph colorings

It is known that, for an oriented hypergraph with (vertex) coloring number $\chi$ and smallest and largest normalized Laplacian eigenvalues $\lambda_1$ and $\lambda_N$, respectively, the inequality $\chi\geq (\lambda_N-\lambda_1)/\min\{\lambda_N-1,1-\lambda_1\}$ holds. We provide necessary conditions for oriented hypergraphs for which this bound is tight. Focusing on $c$-uniform unoriented hypergraphs, we then generalize the bound to the setting of \emph{$d$-proper colorings}: colorings in which no edge contains more than $d$ vertices of the same color. We also adapt our proof techniques to derive analogous spectral bounds for \emph{$d$-improper colorings} of graphs and for edge colorings of hypergraphs. Moreover, for all coloring notions considered, we provide necessary conditions under which the bound is an equality.

math.CO↗

arXiv2024

At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian

For a graph with largest normalized Laplacian eigenvalue $\lambda_N$ and (vertex) coloring number $\chi$, it is known that $\lambda_N\geq \chi/(\chi-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the multiplicity of $\chi/(\chi-1)$. We then describe a family of graphs with largest eigenvalue $\chi/(\chi-1)$. We also study the spectrum of the $1$-sum of two graphs (also known as graph joining or coalescing), with a focus on the maximal eigenvalue. Finally, we give upper bounds on $\lambda_N$ in terms of $\chi$.

math.CO↗

arXiv2023

Maximal colourings for graphs

We consider two different notions of graph colouring, namely, the $t$-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them.

math.CO↗

arXiv2023

Uniform density in matroids, matrices and graphs

We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope contains a point with constant coordinates. As a main application, we derive new spectral, structural and classification results for uniformly dense graphs. In particular, we show that connected regular uniformly dense graphs are $1$-tough and thus contain a (near-)perfect matching. As a second application, we show that strictly uniformly dense real represented matroids can be represented by projection matrices with a constant diagonal and that they are parametrized by a subvariety of the Grassmannian.

math.CO↗

arXiv2023

Petals and Books: The largest Laplacian spectral gap from 1

We prove that, for any connected graph on $N\geq 3$ vertices, the spectral gap from the value $1$ with respect to the normalized Laplacian is at most $1/2$. Moreover, we show that equality is achieved if and only if the graph is either a petal graph (for $N$ odd) or a book graph (for $N$ even). This implies that $(\frac12,\frac32)$ is a maximal gap interval for the normalized Laplacian on connected graphs. This is closely related to the Alon-Boppana bound on regular graphs and a recent result by Kollár and Sarnak on cubic graphs. Our result also provides a sharp bound for the convergence rate of some eigenvalues of the Laplacian on neighborhood graphs.

math.CO↗

arXiv2023

There is no going back: Properties of the non-backtracking Laplacian

We prove new properties of the non-backtracking graph and the non-backtracking Laplacian for graphs. In particular, among other results, we prove that two simple graphs are isomorphic if and only if their corresponding non-backtracking graphs are isomorphic, and we investigate properties of various classes of non-backtracking Laplacian eigenfunctions, such as symmetric and antisymmetric eigenfunctions. Moreover, we introduce and study circularly partite graphs as a generalization of bipartite graphs, and we use this notion to state a sharp upper bound for the spectral gap from $1$. We also investigate the singular values of the non-backtracking Laplacian in relation to independence numbers, and we use them to bound the moduli of the eigenvalues.

math.CO↗

arXiv2023

On Turán numbers for disconnected hypergraphs

We introduce the following simpler variant of the Turán problem: Given integers $n>k>r\geq 2$ and $m\geq 1$, what is the smallest integer $t$ for which there exists an $r$-uniform hypergraph with $n$ vertices, $t$ edges and $m$ connected components such that any $k$-subset of the vertex set contains at least one edge? We prove some general estimates for this quantity and for its limit, normalized by $\binom{n}{r}$, as $n\rightarrow \infty$. Moreover, we give a complete solution of the problem for the particular case when $k=5$, $r=3$ and $m\geq 2$.

math.CO↗

arXiv2023

A measure-theoretic representation of graphs

Inspired by the notion of action convergence in graph limit theory, we introduce a measure-theoretic representation of matrices, and we use it to define a new notion of pseudo-metric on the space of matrices. Moreover, we show that such pseudo-metric is a metric on the subspace of adjacency or Laplacian matrices for graphs. Hence, in particular, we obtain a metric for isomorphism classes of graphs. Additionally, we study how some properties of graphs translate in this measure representation, and we show how our analysis contributes to a simpler understanding of action convergence of graphops.

math.CO↗

arXiv2022

Spectral theory of the non-backtracking Laplacian for graphs

We introduce a non-backtracking Laplace operator for graphs and we investigate its spectral properties. With the use of both theoretical and computational techniques, we show that the spectrum of this operator captures several structural properties of the graph in a more precise way than the classical operators that have been studied so far in the literature, including the non-backtracking matrix.

math.SP↗

arXiv2021

Spectral theory of weighted hypergraphs via tensors

One way to study an hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eingenvalues with some techniques from numerical algebraic geometry.

math.CO↗

arXiv2021

Estimating cellular redundancy in networks of genetic expression

Networks of genetic expression can be modelled by hypergraphs with the additional structure that real coefficients are given to each vertex-edge incidence. The spectra, i.e. the multiset of the eigenvalues, of such hypergraphs, are known to encode structural information of the data. We show how these spectra can be used, in particular, in order to give an estimation of cellular redundancy, a novel measure of gene expression heterogeneity, of the network. We analyze some simulated and real data sets of gene expression for illustrating the new method proposed here.

q-bio.MN↗

arXiv2021

Coloring the normalized Laplacian for oriented hypergraphs

The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a ratio--like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring number.

math.CO↗

arXiv2021

Geometry and symmetry in biochemical reaction systems

Complex systems of intracellular biochemical reactions have a central role in regulating cell identities and functions. Biochemical reaction systems are typically studied using the language and tools of graph theory. However, graph representations only describe pairwise interactions between molecular species, and so are not well suited to modelling complex sets of reactions that may involve numerous reactants and/or products. Here we make use of a recently-developed hypergraph theory of chemical reactions that naturally allows for higher-order interactions to explore the geometry and quantify functional redundancy in biochemical reactions systems. Our results constitute a general theory of automorphisms for oriented hypergraphs and describe the effect of automorphism group structure on hypergraph Laplacian spectra.

math.CO↗

arXiv2021

Random walks and Laplacians on hypergraphs: When do they match?

We develop a general theory of random walks on hypergraphs which includes, as special cases, the different models that are found in literature. In particular, we introduce and analyze general random walk Laplacians for hypergraphs, and we compare them to hypergraph normalized Laplacians that are not necessarily related to random walks, but which are motivated by biological and chemical networks. We show that, although these two classes of Laplacians coincide in the case of graphs, they appear to have important conceptual differences in the general case. We study the spectral properties of both classes, as well as their applications to Coupled Hypergraph Maps: discrete-time dynamical systems that generalize the well-known Coupled Map Lattices on graphs. Our results also show why for some hypergraph Laplacian variants one expects more classical results from (weighted) graphs to generalize directly, while these results must fail for other hypergraph Laplacians.

math.SP↗

arXiv2021

$p$-Laplace Operators for Oriented Hypergraphs

The $p$-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex $p$-Laplacian and a hyperedge $p$-Laplacian are defined for oriented hypergraphs, for all $p\geq 1$. Several spectral properties of these operators are investigated.

math.CO↗