Giusy Monzillo | University of Primorska (original) (raw)

Papers by Giusy Monzillo

arXiv (Cornell University), Apr 18, 2024

The Q-polynomial property is an algebraic property of distance-regular graphs, that was introduce... more The Q-polynomial property is an algebraic property of distance-regular graphs, that was introduced by Delsarte in his study of coding theory. Many distance-regular graphs admit the Q-polynomial property. Only recently the Q-polynomial property has been generalized to graphs that are not necessarily distance-regular. In [21] it was shown that graphs arising from the Hasse diagrams of the so-called attenuated space posets are Q-polynomial. These posets could be viewed as q-analogs of the Hamming posets, which were not studied in [21]. The main goal of this paper is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are Q-polynomial.

arXiv (Cornell University), Apr 5, 2024

Let Γ = Γ(A) denote a simple strongly connected digraph with vertex set X, diameter D, and let {A... more Let Γ = Γ(A) denote a simple strongly connected digraph with vertex set X, diameter D, and let {A 0 , A := A 1 , A 2 ,. .. , A D } denote the set of distance-i matrices of Γ. Let {R i } D i=0 denotes a partition of X × X, where R i = {(x, y) ∈ X × X | (A i) xy = 1} (0 ≤ i ≤ D). The digraph Γ is distance-regular if and only if (X, {R i } D i=0) is a commutative association scheme. In this paper, we describe the combinatorial structure of Γ in the sense of equitable partition, and from it we derive several new algebraic characterizations of such a graph, including the spectral excess theorem for distance-regular digraph. Along the way, we also rediscover all well-known algebraic characterizations of such graphs.

Designs, codes and cryptography, Mar 29, 2024

We provide classification results for translation generalized quadrangles of order less than or e... more We provide classification results for translation generalized quadrangles of order less than or equal to 64, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in PG(3n − 1, 2), for n = 3, 4, and that of the pseudoovals in PG(3n − 1, q), for n = 5, 6, such that one of the associated projective planes is Desarguesian.

arXiv (Cornell University), Mar 1, 2024

Let Γ denote a finite (strongly) connected regular (di)graph with adjacency matrix A. The Hoffman... more Let Γ denote a finite (strongly) connected regular (di)graph with adjacency matrix A. The Hoffman polynomial h(t) of Γ = Γ(A) is the unique polynomial of smallest degree satisfying h(A) = J, where J denotes the all-ones matrix. Let X denote a nonempty finite set. A nonnegative matrix B ∈ Mat X (R) is called λ-doubly stochastic if z∈X (B) yz = z∈X (B) zy = λ for each y ∈ X. In this paper we first show that there exists a polynomial h(t) such that h(B) = J if and only if B is a λ-doubly stochastic irreducible matrix. This result allows us to define the Hoffman polynomial of a λ-doubly stochastic irreducible matrix. Now, let B ∈ Mat X (R) denote a normal irreducible nonnegative matrix, and B = {p(B) | p ∈ C[t]} denote the vector space over C of all polynomials in B. Let us define a 01-matrix A in the following way: (A) xy = 1 if and only if (B) xy > 0 (x, y ∈ X). Let Γ = Γ(A) denote a (di)graph with adjacency matrix A, diameter D, and let A D denote the distance-D matrix of Γ. We show that B is the Bose-Mesner algebra of a commutative D-class association scheme if and only if B is a normal λ-doubly stochastic matrix with D + 1 distinct eigenvalues and A D is a polynomial in B.

arXiv (Cornell University), Aug 30, 2023

Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex... more Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R. Fix a vertex x ∈ X, and define R f = R \ {yz | ∂(x, y) = ∂(x, z)}, where ∂ denotes the path-length distance in Γ. Observe that the graph Γ f = (X, R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x in the sense of Miklavič and Terwilliger [7]. Assume that Γ is a distance-regular graph with classical parameters (D, q, α, β) and diameter D ≥ 4. Recall that q is an integer such that q ∈ {−1, 0}. The purpose of this paper is to study when Γ supports a uniform structure with respect to x. We studied the case q ≤ 1 in [3], and so in this paper we assume q ≥ 2. Let T = T (x) denote the Terwilliger algebra of Γ with respect to x. Under an additional assumption that every irreducible T-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to x, then either α = 0 or α = q, β = q 2 (q D − 1)/(q − 1), and D ≡ 0 (mod 6).

arXiv (Cornell University), May 15, 2023

Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex... more Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R. Fix a vertex x ∈ X, and define R f = R \ {yz | ∂(x, y) = ∂(x, z)}, where ∂ denotes the path-length distance in Γ. Observe that the graph Γ f = (X, R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x. Assume that Γ is a distance-regular graph with classical parameters (D, q, α, β) with q ≤ 1. Recall that q is an integer, which is not equal to 0 or −1. The purpose of this paper is to study when Γ supports a uniform structure with respect to x. The main result of the paper is a complete classification of graphs with classical parameters with q ≤ 1 and D ≥ 4 that support a uniform structure with respect to x.

arXiv (Cornell University), Jul 21, 2023

Let M denote the Bose-Mesner algebra of a commutative d-class association scheme X (not necessari... more Let M denote the Bose-Mesner algebra of a commutative d-class association scheme X (not necessarily symmetric), and Γ denote a (strongly) connected (directed) graph with adjacency matrix A. Under the assumption that A belongs to M, we describe the combinatorial structure of Γ. Moreover, we provide an algebraic-combinatorial characterization of Γ when A generates M. Among else, we show that, if X is a commutative 3-class association scheme that is not an amorphic symmetric scheme, then we can always find a (directed) graph Γ such that the adjacency matrix A of Γ generates the Bose-Mesner algebra M of X.

arXiv (Cornell University), May 5, 2023

Brown et al. provide a representation of a spread of the Tits quadrangle T 2 (O), O an oval of PG... more Brown et al. provide a representation of a spread of the Tits quadrangle T 2 (O), O an oval of PG(2, q), q even, in terms of a certain family of q ovals of PG(2, q). By combining this representation with the Vandendriessche classification of hyperovals in PG(2, 64) and the classification of flocks of the quadratic cone in PG(3, 64), recently given by the authors, in this paper, we classify all the spreads of T 2 (O), O an oval of PG(2, 64), up to equivalence. These complete the classification of spreads of T 2 (O) for q ≤ 64.

arXiv (Cornell University), May 11, 2022

In 2013, van Dam, Martin and Muzychuk constructed a cometric Q− antipodal 4−class association sch... more In 2013, van Dam, Martin and Muzychuk constructed a cometric Q− antipodal 4−class association scheme from a GQ of order (t 2 , t), t odd, which have a hemisystem. In this paper we characterize this scheme by its Krein array. The techniques which are used involve the triple intersection numbers introduced by Coolsaet and Jurišić.

arXiv (Cornell University), May 11, 2022

Penttila and Williford constructed a 4−class association scheme from a generalized quadrangle wit... more Penttila and Williford constructed a 4−class association scheme from a generalized quadrangle with a doubly subtended subquadrangle. We show that an association scheme with appropriate parameters and satisfying some assumption about maximal cliques must be the Penttila-Williford scheme.

arXiv (Cornell University), May 11, 2022

Flocks are an important topic in the field of finite geometry, with many relations with other obj... more Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the * The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3, q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3, 64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2, q), q even, and uses the prior classification of hyperovals in PG(2, 64).

Bulletin of the Malaysian Mathematical Sciences Society

Finite Fields and Their Applications

arXiv (Cornell University), Jan 8, 2023

The research was supported by the Italian National Group for Algebraic and Geometric Structures a... more The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in PG(19, 3). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.

Designs, Codes and Cryptography

Finite Fields and Their Applications

In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-cl... more In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H(3,q^2), q even, with respect to a symplectic polar space W(3,q) embedded in it. In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of PGL(2,q^2), q even, on a non-degenerate conic of PG(2,q^2) extended in PG(2,q^4). Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs.

A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration ... more A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q^n,q^n) and a Laguerre plane of order q^n (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q^-(5,q), non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q^-(5,q) is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q^-(5,q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are...

European Journal of Combinatorics

arXiv (Cornell University), Apr 18, 2024

The Q-polynomial property is an algebraic property of distance-regular graphs, that was introduce... more The Q-polynomial property is an algebraic property of distance-regular graphs, that was introduced by Delsarte in his study of coding theory. Many distance-regular graphs admit the Q-polynomial property. Only recently the Q-polynomial property has been generalized to graphs that are not necessarily distance-regular. In [21] it was shown that graphs arising from the Hasse diagrams of the so-called attenuated space posets are Q-polynomial. These posets could be viewed as q-analogs of the Hamming posets, which were not studied in [21]. The main goal of this paper is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are Q-polynomial.

arXiv (Cornell University), Apr 5, 2024

Let Γ = Γ(A) denote a simple strongly connected digraph with vertex set X, diameter D, and let {A... more Let Γ = Γ(A) denote a simple strongly connected digraph with vertex set X, diameter D, and let {A 0 , A := A 1 , A 2 ,. .. , A D } denote the set of distance-i matrices of Γ. Let {R i } D i=0 denotes a partition of X × X, where R i = {(x, y) ∈ X × X | (A i) xy = 1} (0 ≤ i ≤ D). The digraph Γ is distance-regular if and only if (X, {R i } D i=0) is a commutative association scheme. In this paper, we describe the combinatorial structure of Γ in the sense of equitable partition, and from it we derive several new algebraic characterizations of such a graph, including the spectral excess theorem for distance-regular digraph. Along the way, we also rediscover all well-known algebraic characterizations of such graphs.

Designs, codes and cryptography, Mar 29, 2024

We provide classification results for translation generalized quadrangles of order less than or e... more We provide classification results for translation generalized quadrangles of order less than or equal to 64, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in PG(3n − 1, 2), for n = 3, 4, and that of the pseudoovals in PG(3n − 1, q), for n = 5, 6, such that one of the associated projective planes is Desarguesian.

arXiv (Cornell University), Mar 1, 2024

Let Γ denote a finite (strongly) connected regular (di)graph with adjacency matrix A. The Hoffman... more Let Γ denote a finite (strongly) connected regular (di)graph with adjacency matrix A. The Hoffman polynomial h(t) of Γ = Γ(A) is the unique polynomial of smallest degree satisfying h(A) = J, where J denotes the all-ones matrix. Let X denote a nonempty finite set. A nonnegative matrix B ∈ Mat X (R) is called λ-doubly stochastic if z∈X (B) yz = z∈X (B) zy = λ for each y ∈ X. In this paper we first show that there exists a polynomial h(t) such that h(B) = J if and only if B is a λ-doubly stochastic irreducible matrix. This result allows us to define the Hoffman polynomial of a λ-doubly stochastic irreducible matrix. Now, let B ∈ Mat X (R) denote a normal irreducible nonnegative matrix, and B = {p(B) | p ∈ C[t]} denote the vector space over C of all polynomials in B. Let us define a 01-matrix A in the following way: (A) xy = 1 if and only if (B) xy > 0 (x, y ∈ X). Let Γ = Γ(A) denote a (di)graph with adjacency matrix A, diameter D, and let A D denote the distance-D matrix of Γ. We show that B is the Bose-Mesner algebra of a commutative D-class association scheme if and only if B is a normal λ-doubly stochastic matrix with D + 1 distinct eigenvalues and A D is a polynomial in B.

arXiv (Cornell University), Aug 30, 2023

Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex... more Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R. Fix a vertex x ∈ X, and define R f = R \ {yz | ∂(x, y) = ∂(x, z)}, where ∂ denotes the path-length distance in Γ. Observe that the graph Γ f = (X, R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x in the sense of Miklavič and Terwilliger [7]. Assume that Γ is a distance-regular graph with classical parameters (D, q, α, β) and diameter D ≥ 4. Recall that q is an integer such that q ∈ {−1, 0}. The purpose of this paper is to study when Γ supports a uniform structure with respect to x. We studied the case q ≤ 1 in [3], and so in this paper we assume q ≥ 2. Let T = T (x) denote the Terwilliger algebra of Γ with respect to x. Under an additional assumption that every irreducible T-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to x, then either α = 0 or α = q, β = q 2 (q D − 1)/(q − 1), and D ≡ 0 (mod 6).

arXiv (Cornell University), May 15, 2023

Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex... more Let Γ = (X, R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R. Fix a vertex x ∈ X, and define R f = R \ {yz | ∂(x, y) = ∂(x, z)}, where ∂ denotes the path-length distance in Γ. Observe that the graph Γ f = (X, R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x. Assume that Γ is a distance-regular graph with classical parameters (D, q, α, β) with q ≤ 1. Recall that q is an integer, which is not equal to 0 or −1. The purpose of this paper is to study when Γ supports a uniform structure with respect to x. The main result of the paper is a complete classification of graphs with classical parameters with q ≤ 1 and D ≥ 4 that support a uniform structure with respect to x.

arXiv (Cornell University), Jul 21, 2023

Let M denote the Bose-Mesner algebra of a commutative d-class association scheme X (not necessari... more Let M denote the Bose-Mesner algebra of a commutative d-class association scheme X (not necessarily symmetric), and Γ denote a (strongly) connected (directed) graph with adjacency matrix A. Under the assumption that A belongs to M, we describe the combinatorial structure of Γ. Moreover, we provide an algebraic-combinatorial characterization of Γ when A generates M. Among else, we show that, if X is a commutative 3-class association scheme that is not an amorphic symmetric scheme, then we can always find a (directed) graph Γ such that the adjacency matrix A of Γ generates the Bose-Mesner algebra M of X.

arXiv (Cornell University), May 5, 2023

Brown et al. provide a representation of a spread of the Tits quadrangle T 2 (O), O an oval of PG... more Brown et al. provide a representation of a spread of the Tits quadrangle T 2 (O), O an oval of PG(2, q), q even, in terms of a certain family of q ovals of PG(2, q). By combining this representation with the Vandendriessche classification of hyperovals in PG(2, 64) and the classification of flocks of the quadratic cone in PG(3, 64), recently given by the authors, in this paper, we classify all the spreads of T 2 (O), O an oval of PG(2, 64), up to equivalence. These complete the classification of spreads of T 2 (O) for q ≤ 64.

arXiv (Cornell University), May 11, 2022

In 2013, van Dam, Martin and Muzychuk constructed a cometric Q− antipodal 4−class association sch... more In 2013, van Dam, Martin and Muzychuk constructed a cometric Q− antipodal 4−class association scheme from a GQ of order (t 2 , t), t odd, which have a hemisystem. In this paper we characterize this scheme by its Krein array. The techniques which are used involve the triple intersection numbers introduced by Coolsaet and Jurišić.

arXiv (Cornell University), May 11, 2022

Penttila and Williford constructed a 4−class association scheme from a generalized quadrangle wit... more Penttila and Williford constructed a 4−class association scheme from a generalized quadrangle with a doubly subtended subquadrangle. We show that an association scheme with appropriate parameters and satisfying some assumption about maximal cliques must be the Penttila-Williford scheme.

arXiv (Cornell University), May 11, 2022

Flocks are an important topic in the field of finite geometry, with many relations with other obj... more Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the * The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3, q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3, 64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2, q), q even, and uses the prior classification of hyperovals in PG(2, 64).

Bulletin of the Malaysian Mathematical Sciences Society

Finite Fields and Their Applications

arXiv (Cornell University), Jan 8, 2023

The research was supported by the Italian National Group for Algebraic and Geometric Structures a... more The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in PG(19, 3). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.

Designs, Codes and Cryptography

Finite Fields and Their Applications

In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-cl... more In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H(3,q^2), q even, with respect to a symplectic polar space W(3,q) embedded in it. In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of PGL(2,q^2), q even, on a non-degenerate conic of PG(2,q^2) extended in PG(2,q^4). Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs.

A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration ... more A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q^n,q^n) and a Laguerre plane of order q^n (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q^-(5,q), non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q^-(5,q) is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q^-(5,q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are...

European Journal of Combinatorics