arXiv subjects

Igal Sason

Publications and source records attributed to Igal Sason.

At least 19 recordsLinked to original sources

arXiv2026

Connected irregular cospectral graphs with identical combinatorial invariants and distinct Lov\'{a}sz numbers

For every integer $n \geq 11$, we construct a pair of connected, irregular, nonisomorphic graphs, each on $n$ vertices, that are cospectral with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices and have the same independence number, clique number, chromatic number, and chromatic number of their complements, but have distinct Lov\'{a}sz $\vartheta$-numbers that are independent of $n$. For $n=10$, we first exhibit a regular pair satisfying the analogous conditions. Our construction takes the ordinary join of each member of this fixed regular pair on ten vertices with a complete graph, thereby preserving simultaneous cospectrality with respect to all four matrices, as well as the respective Lov\'{a}sz numbers. We derive exact analytic expressions for these Lov\'{a}sz numbers and prove that they are distinct, without relying on numerical approximations. An exhaustive search shows that no pair of connected, irregular, nonisomorphic graphs on fewer than ten vertices is simultaneously cospectral with respect to the four matrices while also having identical values of the four listed combinatorial invariants. Moreover, the search produces a pair of connected, irregular, nonisomorphic graphs on ten vertices satisfying all these conditions and having distinct Lov\'{a}sz $\vartheta$-numbers. Consequently, the smallest order for which such a pair of connected, irregular graphs exists is exactly 10, and such a pair exists for every $n \geq 10$. This strengthens an earlier existence result for all even orders $n \geq 14$ (Sason, 2024). Beyond its role in bounding the Shannon capacity and other combinatorial graph invariants, the Lov\'{a}sz number thus provides an efficiently computable certificate of nonisomorphism for a family of graph pairs that cannot be distinguished by the four spectra or the four listed invariants.

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arXiv2026

Shannon Capacity and Related Graph Invariants for Lexicographic Products

This paper studies the Shannon capacity of lexicographic products of finite simple graphs, together with the Lov\'{a}sz theta function and the fractional Haemers number. The Shannon capacity is proved to be supermultiplicative under lexicographic products in either order, and these products are compared with the strong product. We explicitly construct three countably infinite families of lexicographic powers based on the Schl\"{a}fli graph, the McLaughlin graph, and its second subconstituent; in each family, pairing each member with its complement yields strict supermultiplicativity and arbitrarily large multiplicative gaps. Bounds and exact-capacity criteria for lexicographic products are derived, and the resulting upper bounds are shown to be incomparable. The capacities of lexicographic products involving Kneser graphs, their complements, and $q$-analogues of Kneser graphs are determined. It is also shown that a lexicographic product with a complete outer factor preserves the Shannon capacity of an arbitrary inner factor. The capacities of iterated lexicographic powers are determined, including those of self-complementary graphs that are vertex-transitive or strongly regular. Elementary, self-contained proofs are also given for three known results: the multiplicativity of the Lov\'{a}sz theta function and the fractional Haemers number under lexicographic products, and the equality of the fractional and ordinary Lov\'{a}sz theta functions. Finally, an open problem concerning the Shannon capacities of lexicographic and strong products is posed.

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arXiv2026

On the transitivity of Gilbert graphs and their complements

The Gilbert graph $\text{Gilbert}(q,n,d)$, which arises naturally in graph theory and coding theory, is the regular graph on $\mathbb{F}_q^n$ in which two vertices are adjacent if their Hamming distance is less than $d$, and it is vertex-transitive. We classify all parameters $(q,n,d)$ for which $\text{Gilbert}(q,n,d)$ is edge-transitive or distance-transitive, and separately classify all parameters for which its complement has these properties. We prove that $\text{Gilbert}(q,n,d)$ is edge-transitive if and only if it is distance-transitive, and that this occurs precisely when $d=2$, $(q,d)=(2,3)$, or $(q,d)=(2,n)$. For the complement graphs, we determine all parameters yielding edge- or distance-transitivity using spectral methods based on Krawtchouk polynomials and the structure of the Hamming association scheme. In contrast to the Gilbert graphs, where the parameter sets corresponding to edge- and distance-transitivity coincide, we show that for their complements the set of parameters yielding distance-transitivity is strictly contained in the set yielding edge-transitivity. As an application, we compute the exact values of the Lov\'{a}sz $\vartheta$-function of Gilbert graphs, as well as of their complements, in all cases where either one of them is edge-transitive.

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arXiv2026

The Lov\'{a}sz Local Lemma: Foundations and Applications

The Lov\'{a}sz Local Lemma (LLL) is a central tool in probabilistic combinatorics, providing a sufficient condition under which a finite collection of undesirable events with limited dependencies can be simultaneously avoided with positive probability. This paper offers a self-contained expository treatment of the lemma and its strengthened versions, emphasizing mathematical foundations, conceptual clarity, and applications. We begin with a pedagogically motivated proof of the LLL based entirely on unconditional probability inequalities. Particular attention is given to the symmetric form of the lemma and several subsequent strengthenings. We also discuss a variety of classical applications of both the symmetric and asymmetric forms of the LLL in combinatorics and graph theory, including bounds for the edge-disjoint paths problem, satisfiability of Boolean formulas in conjunctive normal form, lower bounds on diagonal and off-diagonal Ramsey numbers, hypergraph coloring results, structural properties of directed graphs, and acyclic graph colorings. Additional observations and refinements are provided throughout. We also introduce the algorithmic framework of Moser and Tardos, highlighting its constructive counterpart to the LLL, together with an introduction to the entropy-compression principle. The lopsided LLL, a refinement of the LLL, is presented along with an application to the Latin transversal problem. We further discuss the cluster-expansion lemma and its relation to the LLL, and present an alternative treatment of the Latin transversal problem from the cluster-expansion perspective that yields an improved result. The exposition concludes with a high-level overview of the iterated LLL, also known as the semi-random method.

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arXiv2025

Advances in the Shannon Capacity of Graphs

We derive exact values and new bounds for the Shannon capacity of two families of graphs: the $q$-Kneser graphs and the tadpole graphs. We also construct a countably infinite family of connected graphs whose Shannon capacity is not attained by the independence number of any finite strong power. Building on recent work of Schrijver, we establish sufficient conditions under which the Shannon capacity of a polynomial in graphs, formed via disjoint unions and strong products, equals the corresponding polynomial of the individual capacities, thereby reducing the evaluation of such capacities to that of their components. Finally, we prove an inequality relating the Shannon capacities of the strong product of graphs and their disjoint union, which yields alternative proofs of several known bounds as well as new tightness conditions. In addition to contributing to the computation of the Shannon capacity of graphs, this paper is intended to serve as an accessible entry point to those wishing to work in this area.

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arXiv2025

Counting Graph Homomorphisms in Bipartite Settings

This paper studies the problem of counting homomorphisms from a bipartite source graph to a bipartite target graph. An exact formula is first derived for the number of homomorphisms from a complete bipartite graph to a general bipartite graph. Although exact, its evaluation is typically computationally intensive, and a computationally tractable combinatorial lower bound is derived. When the target graph contains no 4-cycles, the lower bound simplifies and becomes exact. Two additional lower bounds on the number of homomorphisms from a complete bipartite graph to an arbitrary bipartite graph are derived using properties of Shannon entropy. The first depends only on the sizes of the partite sets in the source and target graphs, together with the edge density of the target graph. The second further incorporates the degree profiles of the partite sets of the target graph, thereby strengthening the first bound. Both entropy-based bounds improve upon the inequality implied by the validity of Sidorenko's conjecture for complete bipartite source graphs. The lower bounds for complete bipartite source graphs are combined with new auxiliary results to derive general lower bounds on homomorphism counts between arbitrary bipartite graphs. Furthermore, a known reverse Sidorenko inequality is employed to derive a corresponding upper bound. This upper bound is attained when the source graph is a disjoint union of complete bipartite graphs, and admits a simple closed-form expression when the target graph contains no 4-cycles. Numerical results compare the new computationally tractable bounds with exact homomorphism counts in cases where exact computation is feasible.

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arXiv2025

An example showing that Schrijver's $\vartheta$-function need not upper bound the Shannon capacity of a graph

This letter addresses an open question concerning a variant of the Lov\'{a}sz $\vartheta$ function, which was introduced by Schrijver and independently by McEliece et al. (1978). The question of whether this variant provides an upper bound on the Shannon capacity of a graph was explicitly stated by Bi and Tang (2019). This letter presents an explicit example of a Tanner graph on 32 vertices, which shows that, in contrast to the Lov\'{a}sz $\vartheta$ function, this variant does not necessarily upper bound the Shannon capacity of a graph. The example, previously outlined by the author in a recent paper (2024), is presented here in full detail, making it easy to follow and verify. By resolving this question, the note clarifies a subtle but significant distinction between these two closely related graph invariants.

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arXiv2025

On Strongly Regular Graphs and the Friendship Theorem

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erd\H{o}s, R\'{e}nyi, and S\'{o}s (1966). The proof relies on a closed-form expression for the Lov\'{a}sz $\vartheta$-function of strongly regular graphs, recently derived by the author. Additionally, the paper considers some known extensions of the theorem, offering discussions that provide insights into the friendship theorem, one of its extensions, and the proposed proof. Leveraging the closed-form expression for the Lov\'{a}sz $\vartheta$-function of strongly regular graphs, the paper further establishes new necessary conditions for a strongly regular graph to be a spanning or induced subgraph of another strongly regular graph. In the case of induced subgraphs, the analysis also incorporates a property of graph energies. Some of these results are extended to regular graphs and their subgraphs.

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arXiv2025

On H-Intersecting Graph Families and Counting of Homomorphisms

This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H. Such families are referred to as H-intersecting graph families. The bound is derived using the combinatorial version of Shearer's lemma, and it forms a nontrivial extension of the bound derived by Chung, Graham, Frankl, and Shearer (1986), where H is specialized to a triangle. The derived bound is expressed in terms of the chromatic number of H, while a relaxed version, formulated using the Lov\'{a}sz $\vartheta$-function of the complement of H, offers reduced computational complexity. Additionally, a probabilistic version of Shearer's lemma, combined with properties of the Shannon entropy, are employed to establish bounds related to the enumeration of graph homomorphisms, providing further insights into the interplay between combinatorial structures and information-theoretic principles.

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arXiv2024

On Spectral Graph Determination

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph's spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights.

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arXiv2024

Observations on Graph Invariants with the Lovász $\vartheta$-Function

This paper delves into three research directions, leveraging the Lovász $\vartheta$-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that $n \geq 14$, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on $n$ vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász $\vartheta$-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the $\vartheta$-function by Schrijver and the identical function by McEliece et al. It shows, by a counterexample, that the $\vartheta$-function variant by Schrijver does not possess the property of the Lovász $\vartheta$-function of forming an upper bound on the Shannon capacity of a graph.

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arXiv2023

Observations on the Lovász $θ$-Function, Graph Capacity, Eigenvalues, and Strong Products

This paper provides new observations on the Lovász $θ$-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász $θ$-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász $θ$-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a $k$-fold strong power of a regular graph is compared to the Alon--Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in $k$). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász $θ$-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.

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arXiv2022

Information Inequalities via Submodularity and a Problem in Extremal Graph Theory

The present paper offers, in its first part, a unified approach for the derivation of families of inequalities for set functions which satisfy sub/supermodularity properties. It applies this approach for the derivation of information inequalities with Shannon information measures. Connections of the considered approach to a generalized version of Shearer's lemma, and other related results in the literature are considered. Some of the derived information inequalities are new, and also known results (such as a generalized version of Han's inequality) are reproduced in a simple and unified way. In its second part, this paper applies the generalized Han's inequality to analyze a problem in extremal graph theory. This problem is motivated and analyzed from the perspective of information theory, and the analysis leads to generalized and refined bounds. The two parts of this paper are meant to be independently accessible to the reader.

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arXiv2021

Entropy-Based Proofs of Combinatorial Results on Bipartite Graphs

This work considers new entropy-based proofs of some known, or otherwise refined, combinatorial bounds for bipartite graphs. These include upper bounds on the number of the independent sets, lower bounds on the minimal number of colors in constrained edge coloring, and lower bounds on the number of walks of a given length in bipartite graphs. The proofs of these combinatorial results rely on basic properties of the Shannon entropy.

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arXiv2021

On Two-Stage Guessing

Stationary memoryless sources produce two correlated random sequences $X^n$ and $Y^n$. A guesser seeks to recover $X^n$ in two stages, by first guessing $Y^n$ and then $X^n$. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in $n$) of any positive $ρ$-th moment of the total number of guesses when $Y^n$ is obtained by applying a deterministic function $f$ component-wise to $X^n$. We prove that, depending on $f$, the least exponential growth rate in the two-stage setup is lower than when guessing $X^n$ directly. We further propose a simple Huffman code-based construction of a function $f$ that is a viable candidate for the minimization of the least exponential growth rate in the two-stage guessing setup. (2) We characterize the least achievable exponential growth rate of the $ρ$-th moment of the total number of guesses required to recover $X^n$ when Stage 1 need not end with a correct guess of $Y^n$ and without assumptions on the stationary memoryless sources producing $X^n$ and $Y^n$.

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arXiv2021

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn's information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but it may be irregular in the other, the extended entropy-based proof technique yields the same bound that was conjectured by Kahn (2001) and proved by Sah et al. (2019).

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arXiv2021

On Strong Data-Processing and Majorization Inequalities with Applications to Coding Problems

This work provides data-processing and majorization inequalities for $f$-divergences, and it considers some of their applications to coding problems. This work also provides tight bounds on the Rényi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one, and their derivation is based on majorization and the Schur-concavity of the Rényi entropy. One application of the $f$-divergence inequalities refers to the performance analysis of list decoding with either fixed or variable list sizes; some earlier bounds on the list decoding error probability are reproduced in a unified way, and new bounds are obtained and exemplified numerically. Another application is related to a study of the quality of approximating a probability mass function, which is induced by the leaves of a Tunstall tree, by an equiprobable distribution. The compression rates of finite-length Tunstall codes are further analyzed for asserting their closeness to the Shannon entropy of a memoryless and stationary discrete source. In view of the tight bounds for the Rényi entropy and the work by Campbell, non-asymptotic bounds are derived for lossless data compression of discrete memoryless sources.

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arXiv2020

Some Useful Integral Representations for Information-Theoretic Analyses

This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive noninteger real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, $n$, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over $n$ dimensions, as needed in the direct calculation of the desired moment.

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