On Hoffman polynomials of λ\lambda-doubly stochastic irreducible matrices and commutative association schemes

Let $\Gamma$ denote a finite (strongly) connected regular (di)graph with adjacency matrix $A$. The {\em Hoffman polynomial} $h(t)$ of $\Gamma=\Gamma(A)$ 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\in{\mbox{Mat}}_X({\mathbb R}) is called {\em $\lambda$-doubly stochastic} if $\sum_{z\in X} (B)_{yz}=\sum_{z\in X} (B)_{zy}=\lambda$ for each $y\in X$. 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 $\lambda$-doubly stochastic irreducible matrix. This result allows us to define the Hoffman polynomial of a $\lambda$-doubly stochastic irreducible matrix. Now, let B\in{\mbox{Mat}}_X({\mathbb R}) denote a normal irreducible nonnegative matrix, and ${\cal B}=\{p(B)\mid p\in{\mathbb{C}}[t]\}$ denote the vector space over ${\mathbb{C}}$ of all polynomials in $B$. Let us define a $01$-matrix $\widehat{A}$ in the following way: $(\widehat{A})_{xy}=1$ if and only if $(B)_{xy}>0$ $(x,y\in X)$. Let $\Gamma=\Gamma(\widehat{A})$ denote a (di)graph with adjacency matrix $\widehat{A}$, diameter $D$, and let $A_D$ denote the distance-$D$ matrix of $\Gamma$. We show that ${\cal B}$ is the Bose--Mesner algebra of a commutative $D$-class association scheme if and only if $B$ is a normal $\lambda$-doubly stochastic matrix with $D+1$ distinct eigenvalues and $A_D$ is a polynomial in $B$

On combinatorial structure and algebraic characterizations of distance-regular digraphs

Let $\Gamma=\Gamma(A)$ denote a simple strongly connected digraph with vertex set $X$, diameter $D$, and let $\{A_0,A:=A_1,A_2,\ldots,A_D\}$ denote the set of distance-$i$ matrices of $\Gamma$. Let $\{R_i\}_{i=0}^D$ denote a partition of $X\times X$, where $R_i=\{(x,y)\in X\times X\mid (A_i)_{xy}=1\}$ $(0\le i\le D)$. The digraph $\Gamma$ is distance-regular if and only if $(X,\{R_i\}_{i=0}^D)$ is a commutative association scheme. In this paper, we describe the combinatorial structure of $\Gamma$ 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.Comment: arXiv admin note: text overlap with arXiv:2403.0065

Reconstructing a generalized quadrangle from the Penttila–Williford 4-class association scheme

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

On the isomorphism of certain primitive Q-polynomial not P-polynomial association schemes

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,q2), 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,q2), q even, on a non-degenerate conic of PG(2,q2) extended in PG(2,q4). 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

On commutative association schemes and associated (directed) graphs

Let ${\cal M}$ denote the Bose--Mesner algebra of a commutative $d$-class association scheme ${\mathfrak X}$ (not necessarily symmetric), and $\Gamma$ denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under the assumption that $A$ belongs to ${\cal M}$, in this paper, we describe the combinatorial structure of $\Gamma$. Among else, we show that, if ${\mathfrak X}$ is a commutative $3$-class association scheme that is not an amorphic symmetric scheme, then we can always find a (directed) graph $\Gamma$ such that the adjacency matrix $A$ of $\Gamma$ generates the Bose--Mesner algebra ${\cal M}$ of ${\mathfrak X}$

The association scheme on the set of flags of a finite generalized quadrangle

In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is showed that an association scheme with appropriate parameters must arise from the flags of a generalized quadrangle. The same is done for one of its 4-class symmetric fusion

Classification of flocks of the quadratic cone in PG(3,64)

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 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)

ESTUDO DA MORFOLOGIA URBANA DE UM BAIRRO-JARDIM Jardim São Bento, Casa Verde, São Paulo

The history of São Paulo presents significant gaps regarding the development of its neighborhoods, therefore, the objective of the text is to present the historiographical reading of Jardim São Bento, in the north zone of the municipality. The neighborhood is part of a set of allotments implemented in the first half of the 20th century, inspired by the precepts of the garden city and which were used as a means of attraction and residential settlement for a specific layer of the population, the São Paulo elite. The research methodology is based on historiography and makes use of two approaches that are close to the English school of urban morphology: the reconstitution of the historical context of the period of conformation of the urban fragment and the analysis of the urban form, contemplating the questions referring to the implantation in the urban land from the layout of roads and blocks, the division of lots and the occupation of buildings within the lots.   Keywords: urban morphology; garden district; Jardim São Bento; urban form.A história de São Paulo apresenta lacunas significativas sobre o desenvolvimento de seus bairros, portanto, o objetivo do texto é apresentar a leitura historiográfica do Jardim São Bento, na zona norte do município. O bairro faz parte de um conjunto de loteamentos implantados na primeira metade do século XX, inspirados nos preceitos de cidade-jardim e que foram utilizados como meio de atração e assentamento residencial de uma camada específica da população, a elite paulistana. A metodologia de pesquisa é de base historiográfica e faz uso de duas abordagens que se aproximam da escola inglesa de morfologia urbana: a reconstituição do contexto histórico do período de conformação do fragmento urbano e a análise da forma urbana, contemplando as questões referentes à implantação no solo urbano a partir do traçado das vias e quadras, a divisão dos lotes e a ocupação das edificações dentro dos lotes. Palavras chave: morfologia urbana, bairro-jardim, Jardim São Bento, forma urbana.Peer Reviewe

Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

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 (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 introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of Q^-(5,q) can be analysed from this viewpoint