One of the main aims of the paper is to develop the mesh geometry technique for corank-two edge-bipartite graphs Δ with $n+2\ge 3$ vertices, and the mesh algorithms introduced in [30], [33] and successfully studied in our recent article [42]. We introduce and study the concept of a self-duality of mesh geometries $\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ viewed as ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-mesh translation quivers. We show how self-dualities of mesh geometries $\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ and the mesh geometry technique is applied to an affirmative algorithmic solution of so called Horn-Sergeichuk type problem [9, Problem 4.3] on the self-congruency of square integer matrices $A\in {\mathbb{M}}_{n+2}(\mathbb{Z})$, for the class of non-symmetric Gram matrices $A={\check{G}}_{\mathrm{\Delta }}$ of corank-two loop-free edge-bipartite graphs Δ, with $n+2\le 6$ vertices. More precisely, we show that each of the mesh geometries $\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ is self-dual, we construct its dual form $\mathrm{\Gamma }{({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})}^{op}=\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }}^{-1})$ isomorphic with $\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$, and we construct a canonical self-duality isomorphism $f_{\mathrm{\Delta }}:\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})\to \mathrm{\Gamma }{({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})}^{op}$ of mesh translation quivers. Using the self-duality $f_{\mathrm{\Delta }}$ we construct combinatorial algorithms such that, given a square Gram matrix $A={\check{G}}_{\mathrm{\Delta }}\in {\mathbb{M}}_{n+2}(\mathbb{Z})$ of Δ lying in this class, they are able to compute a $\mathbb{Z}$-invertible matrix $B\in {\mathbb{M}}_{n+2}(\mathbb{Z})$ that coincide with its inverse $B^{-1}$ and defines the congruence of A with $A^{tr}$, i.e., the equation $A^{tr}=B^{tr}\cdot A\cdot B$ is satisfied.

An idea of our solution is outlined in Section 8 of our recent article [42], where among others two of our 13 algorithms solving the problem are constructed. The remaining 11 algorithms are constructed in the present article. We do it by means of the structure of the standard self-dual ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-mesh translation quiver $\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ (called a geometry) canonically associated with Δ, consisting of ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-meshes of ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-orbits ${\mathcal{O}}_{\mathrm{\Delta }}(w)$ of vectors $w\in {\hat{\mathcal{R}}}_{\mathrm{\Delta }}\subseteq {\mathbb{Z}}^{n+2}$, where ${\mathrm{\Phi }}_{\mathrm{\Delta }}:{\mathbb{Z}}^{n+2}\to {\mathbb{Z}}^{n+2}$ is the Coxeter transformation of Δ. We construct in the paper such self-dual ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-mesh geometry $\mathrm{\Gamma }({\hat{\mathcal{R}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$, for each of the corank-two loop-free edge-bipartite graphs Δ, with $n+2\le 6$ vertices.

该论文的主要目标之一是开发用于 corank-two 边二部图 Δ 的网格几何技术，其中 $n+2\ge 3$顶点，以及在 [30]、[33] 中介绍并在我们最近的文章 [42] 中成功研究的网格算法。我们介绍并研究了网格几何的自对偶性概念$\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ 被视为 ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-mesh 翻译颤抖。我们展示了网格几何的自对偶性$\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ 并且将网格几何技术应用于所谓的 Horn-Sergeichuk 类型问题 [9, 问题 4.3] 的关于平方整数矩阵自相关性的肯定算法解决方案 $\mathrm{一种}\in {\mathbb{米}}_{n+2}(\mathbb{Z})$, 对于非对称 Gram 矩阵类 $\mathrm{一种}={\check{G}}_{\mathrm{\Delta }}$ corank-2 无环边二部图 Δ，与 $n+2\le 6$顶点。更准确地说，我们展示了每个网格几何形状$\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ 是自对偶的，我们构造它的对偶形式 $\mathrm{\Gamma }{({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})}^{○磷}=\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }}^{-1})$ 同构 $\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$，我们构造了一个规范的自对偶同构 $F_{\mathrm{\Delta }}：\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})\to \mathrm{\Gamma }{({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})}^{○磷}$网格平移颤动。使用自我二元性$F_{\mathrm{\Delta }}$ 我们构造组合算法，使得，给定一个平方 Gram 矩阵 $\mathrm{一种}={\check{G}}_{\mathrm{\Delta }}\in {\mathbb{米}}_{n+2}(\mathbb{Z})$ Δ 在这个类中，他们能够计算出 $\mathbb{Z}$-可逆矩阵 $乙\in {\mathbb{米}}_{n+2}(\mathbb{Z})$ 与它的逆重合 ${乙}^{-1}$并限定的一致性阿与${\mathrm{一种}}^{吨r}$, 即方程 ${\mathrm{一种}}^{吨r}={乙}^{吨r}\cdot \mathrm{一种}\cdot 乙$ 很满意。

我们最近的文章 [42] 的第 8 节概述了我们的解决方案的一个想法，其中构建了我们解决问题的 13 种算法中的两种。其余 11 种算法在本文中构建。我们通过标准自对偶的结构来做${\mathrm{\Phi }}_{\mathrm{\Delta }}$-网状翻译箭袋 $\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$ （称为几何）规范地与 Δ 相关联，包括 ${\mathrm{\Phi }}_{\mathrm{\Delta }}$- 网格 ${\mathrm{\Phi }}_{\mathrm{\Delta }}$-轨道 ${\mathcal{哦}}_{\mathrm{\Delta }}(瓦)$ 向量 $瓦\in {\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }}\subseteq {\mathbb{Z}}^{n+2}$， 在哪里 ${\mathrm{\Phi }}_{\mathrm{\Delta }}：{\mathbb{Z}}^{n+2}\to {\mathbb{Z}}^{n+2}$是 Δ 的 Coxeter 变换。我们在论文中构造了这样的自对偶${\mathrm{\Phi }}_{\mathrm{\Delta }}$-网格几何 $\mathrm{\Gamma }({\widehat{\mathcal{电阻}}}_{\mathrm{\Delta }},{\mathrm{\Phi }}_{\mathrm{\Delta }})$，对于每一个 corank-two 无环边二部图 Δ，有 $n+2\le 6$ 顶点。