Summary: We continue the Coxeter spectral study of finite positive edge-bipartite signed (multi)graphs \(\Delta\) (bigraphs, for short), with \(n\geq 2\) vertices started in [the author, SIAM J. Discrete Math. 27, No. 2, 827–854 (2013; Zbl 1272.05072)] and developed in [the author, Linear Algebra Appl. 557, 105–133 (2018; Zbl 1396.05049)]. We do it by means of the non-symmetric Gram matrix \(\check{G}_{\Delta}\in \mathbb{M}_n(\mathbb{Z})\) defining \(\Delta\), its Gram quadratic form \(q_{\Delta}:\mathbb{Z}^n\to\mathbb{Z}\), \(v\mapsto v\cdot\check{G}_{\Delta}\cdot v^{tr}\) (that is positive definite, by definition), the complex spectrum \(\operatorname{specc}_{\Delta}\subset\mathcal{S}^1:=\{z\in\mathbb{C},|z|=1\}\) of the Coxeter matrix \(\operatorname{Cox}_{\Delta}:=-\check{G}_{\Delta}\cdot \check{G}_{\Delta}^{-tr}\in\mathbb{M}_n(\mathbb{Z})\), called the Coxeter spectrum of \(\Delta\), and the Coxeter polynomial \(\operatorname{cox}_{\Delta}(t):=\det(t\cdot E-\operatorname{Cox}_{\Delta})\in\mathbb{Z} [t]\). One of the aims of the Coxeter spectral analysis is to classify the connected bigraphs \(\Delta\) with \(n\geq 2\) vertices up to the \(\ell\)-weak Gram \(\mathbb{Z}\)-congruence \(\Delta\sim_{\ell\mathbb{Z}} \Delta^\prime\) and up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta\approx_{\mathbb{Z}}\Delta^\prime\), where \(\Delta\sim_{\ell\mathbb{Z}}\Delta^\prime\) (resp. \(\Delta\approx_{\mathbb{Z}}\Delta^\prime)\) means that \(\det \check{G}_{\Delta}=\det \check{G}_{\Delta^\prime}\) and \(G_{\Delta^\prime}=B^{tr}\cdot G_{\Delta}\cdot B\) (resp. \(\check{G}_{\Delta^\prime}=B^{tr}\cdot \check{G}_{\Delta}\cdot B)\), for some \(B\in\mathbb{M}_n(\mathbb{Z})\) with \(\det B=\pm 1\), where \(G_{\Delta}:=\frac{1}{2}[\check{G}_{\Delta}+\check{G}_{\Delta}^{tr}]\in \mathbb{M}_n(\frac{1}{2}\mathbb{Z})\).
Here we study connected signed simple graphs \(\Delta\), with \(n\geq 2\) vertices, that are positive, i.e., the symmetric Gram matrix \(G_{\Delta}\in\mathbb{M}_n(\frac{1}{2}\mathbb{Z})\) of \(\Delta\) is positive definite. It is known that every such a signed graph is \(\ell\)-weak Gram \(\mathbb{Z}\)-congruent with a unique simply laced Dynkin graph \(\operatorname{Dyn}_{\Delta} \in\{\mathbb{A}_n,\mathbb{D}_n,n\geq 4,\mathbb{E}_6, \mathbb{E}_7,\mathbb{E}_8\}\), called the Dynkin type of \(\Delta\). A classification up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta\approx_{\mathbb{Z}}\Delta^\prime\) is still an open problem and only partial results are known. In this paper, we obtain such a classification for the positive signed simple graphs \(\Delta\) of Dynkin type \(\mathbb{D}_n\) by means of the family of the signed graphs \(\mathcal{D}_n^{(1)}=\mathbb{D}_n, \mathcal{D}_n^{(2)},\dots,\mathcal{D}_n^{(r_n)}\) constructed in Section 2, for any \(n\geq 4\), where \(r_n=\llcorner n/2\lrcorner\). More precisely, we prove that any connected signed simple graph of Dynkin type \(\mathbb{D}_n\), with \(n\geq 4\) vertices, is strongly \(\mathbb{Z}\)-congruent with a signed graph \(\mathcal{D}_n^{(s)}\), for some \(s\leq r_n\), and its Coxeter polynomial \(\operatorname{cox}_{\Delta}(t)\) is of the form \((t^s+1)(t^{n-s}+1)\). We do it by a matrix morsification type reduction to the classification of the conjugacy classes in the integral orthogonal group \(\operatorname{O}(n, \mathbb{Z})\) of the integer orthogonal matrices \(C\), with \(\det(E-C)=4\).