Showing papers in "Journal of Geometry in 2022"

Twisted cubic and plane-line incidence matrix in PG(3,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {PG}(3

Abstract: The point-plane, the point-line, and the plane-line incidence matrices of $$\mathrm {PG}(3,q)$$ are of interest in combinatorics, finite geometry, graph theory and group theory. Some of the properties of these matrices and their submatrices are related with the interplay of orbits of points, lines and planes under the action of subgroups of $$\mathrm {PGL}(4,q)$$ . A remarkable particular case is the subgroup $$G\cong \mathrm {PGL}(2,q)$$ , viewed as the stabilizer group of the twisted cubic $$\mathscr {C}$$ . For this case, the study of the point-plane incidence matrix, initiated by D. Bartoli and the present authors, has attracted attention as being related to submatrices with useful applications in coding theory for the construction of multiple covering codes. In this paper, we extend our investigation to the plane-line incidence matrix apart from just one class of the line orbits, named $$\mathcal {O}_6$$ in the literature. For all $$q\ge 2$$ , in each submatrix, the numbers of lines in any plane and planes through any line are obtained. As a tool for the present investigation, we use submatrices of incidences arising from orbits of planes and unions of line orbits, including $$\mathcal {O}_6$$ . For each such submatrix we determine the total number of lines from the union in any plane and the average number of planes from the orbit through any line of the union.

Twisted cubic and plane-line incidence matrix in $$\mathrm {PG}(3,q)$$

Abstract: The point-plane, the point-line, and the plane-line incidence matrices of $$\mathrm {PG}(3,q)$$ are of interest in combinatorics, finite geometry, graph theory and group theory. Some of the properties of these matrices and their submatrices are related with the interplay of orbits of points, lines and planes under the action of subgroups of $$\mathrm {PGL}(4,q)$$ . A remarkable particular case is the subgroup $$G\cong \mathrm {PGL}(2,q)$$ , viewed as the stabilizer group of the twisted cubic $$\mathscr {C}$$ . For this case, the study of the point-plane incidence matrix, initiated by D. Bartoli and the present authors, has attracted attention as being related to submatrices with useful applications in coding theory for the construction of multiple covering codes. In this paper, we extend our investigation to the plane-line incidence matrix apart from just one class of the line orbits, named $$\mathcal {O}_6$$ in the literature. For all $$q\ge 2$$ , in each submatrix, the numbers of lines in any plane and planes through any line are obtained. As a tool for the present investigation, we use submatrices of incidences arising from orbits of planes and unions of line orbits, including $$\mathcal {O}_6$$ . For each such submatrix we determine the total number of lines from the union in any plane and the average number of planes from the orbit through any line of the union.

Notes on flat fronts in hyperbolic space

Abstract: Abstract We give a short introduction to discrete flat fronts in hyperbolic space and prove that any discrete flat front in the mixed area sense admits a Weierstrass representation.

Remarks on the quadratic orthogonal bisectional curvature

Abstract: We exhibit a curious link between the Quadratic Orthogonal Bisectional Curvature, combinatorics, and distance geometry. The Weitzenb\"ock curvature operator, acting on real (1,1)--forms, is realized as the Dirichlet energy of a finite graph, weighted by a matrix of the curvature. These results also illuminate the difference in the nature of the Quadratic Orthogonal Bisectional Curvature and the Real Bisectional Curvature.

Non-geometric cospectral mates of line graphs with a linear representation

Abstract: For an incidence geometry $${\mathcal G}= ({\mathcal P}, {\mathcal L}, {\text {I}})$$ with a linear representation $${\mathcal T}_2^*({\mathcal K})$$ , we apply WQH switching to construct a non-geometric graph $$\Gamma '$$ cospectral with the line graph $$\Gamma $$ of $${\mathcal G}$$ . As an application, we show that for $$h \ge 2$$ and $$0< m < h$$ , there are strongly regular graphs with parameters $$(v, k, \lambda , \mu ) = (2^{2\,h} (2^{m+h}+2^m-2^h), 2^h (2^h+1)(2^m-1), 2^h (2^{m+1}-3), 2^h (2^m-1))$$ which are not point graphs of partial geometries of order $$(s,t,\alpha ) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1)$$ .

New bounds on the existence of $$(n_{5})$$ and $$(n_{6})$$ configurations: the Grünbaum Calculus revisited

Abstract: The “Grünbaum Incidence Calculus” is the common name of a collection of operations introduced by Branko Grünbaum to produce new \((n_{4})\) configurations from various input configurations. In a previous paper, we generalized two of these operations to produce operations on arbitrary \((n_k)\) configurations, and we showed that for each k, there exists an integer \(N_{k}\) such that for all \(n \ge N_{k}\), there exists at least one \((n_{k})\) configuration, with current records \(N_{5}\le 576\) and \(N_{6}\le 7350\). In this paper, we further extend the Grünbaum Calculus; using these operations, as well as a collection of previously known and novel ad hoc constructions, we refine the bounds for \(k = 5\) and \(k = 6\). Namely, we show that \(N_5 \le 166\) and \(N_{6}\le 585\).

Finite subunitals of the Hermitian unitals

Abstract: Abstract Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a hermitian one (i.e., it originates from a separable field extension).

On a strengthening of the Blaschke–Leichtweiss theorem

Abstract: The Blaschke–Leichtweiss theorem (Leichtweiss in Abh Math Semin Univ Hambg 75:257–284, 2005) states that the smallest area convex domain of constant width w in the 2-dimensional spherical space $${\mathbb {S}}^2$$ is the spherical Reuleaux triangle for all $$0

Some characterizations of quasi Yamabe solitons

Abstract: In this article, we have proved some results in connection with the potential vector field having finite global norm in quasi Yamabe soliton. We have derived some criteria for the potential vector field on the non-positive Ricci curvature of the quasi Yamabe soliton. Also, a necessary condition for a compact quasi Yamabe soliton has been formulated. We further showed that if the potential vector field has a finite global norm in a complete non-trivial, non-compact quasi Yamabe soliton with finite volume, then the scalar curvature becomes constant and the soliton reduces to a Yamabe soliton.