--- title: 'Legendre Graph: Geometry & Quadratic Symbols' url: https://www.emergentmind.com/topics/legendre-graph type: topic --- # Legendre Graph: Geometry & Quadratic Symbols Legendre graph is a term used in several distinct senses in current mathematical literature. In symplectic geometry, it denotes the graph \(L_\psi=\{(q,p)\in T^*Q: p=d\psi(q)\}\) of the differential of a smooth strictly convex potential. In arithmetic graph theory, a natural usage is for a Paley-type graph whose adjacency is determined by the Legendre symbol, or more generally by a primitive quadratic Dirichlet character. In graph labeling, it denotes an auxiliary graph \(L_n^k(f,p)\) whose edges are selected according to the Legendre symbol of sums of vertex labels. These meanings share the adjective “Legendre,” but they arise from different structures—Legendre duality in geometry and the Legendre symbol in number theory—and they are not interchangeable [2512.19409] [2212.02005] [2509.09528]. ## 1. Principal meanings and scope The current literature supports three main uses of the term. | Context | Object | Defining feature | |---|---|---| | Symplectic geometry | \(L_\psi\subset T^*Q\) | \(p=d\psi(q)\) | | Quadratic-character graph theory | Paley-type circulant graph | Adjacency from \(\chi_\Delta(v-u)=1\) | | Graph labeling | \(L_n^k(f,p)\) | \(\left(\frac{f(a)+f(b)}{p}\right)=k\) | In the geometric usage, the object is literally a graph in the sense of the graph of a differential. In the arithmetic and combinatorial usages, it is a graph-theoretic object whose edges are decided by quadratic-residue data. A common misconception is to treat “Legendre graph” as a standardized single notion. The literature instead shows a terminological split: one branch is geometric and lives in cotangent bundles, while the other is discrete and number-theoretic [2512.19409] [2212.02005] [2509.09528]. ## 2. Legendre graphs in symplectic geometry On an \(n\)-dimensional smooth manifold \(Q\), with a smooth potential \(\psi:Q\to\mathbb{R}\), the Legendre graph is defined by \[ L_\psi := \{(q,p)\in T^*Q : p = d\psi(q)\}. \] In local coordinates \((q^1,\dots,q^n)\), this is \[ p_i=\frac{\partial \psi}{\partial q^i}, \qquad i=1,\dots,n. \] The same construction is presented as a duality of Legendre type: \(q\) are primal coordinates, \(p\) are dual coordinates, and the duality relation is \(p=d\psi(q)\). In the information-geometric formulation, the parameter manifold is denoted by \(\Xi\), a dualistic model \((\mathcal P,\psi)\) satisfies \(\eta(\theta)=\nabla\psi(\theta)\), and the corresponding Legendre graph is \[ L_\psi=\{(\theta,\eta)\in T^*Q:\eta=d\psi(\theta)\}. \] Thus the graph encodes the statement that dual coordinates are exact derivatives of a potential rather than independent variables [2512.19409]. The cotangent bundle \(T^*Q\) carries the canonical \(1\)-form \[ \alpha=p_i\,dq^i \] and the canonical symplectic form \[ \omega=-d\alpha=dq^i\wedge dp_i. \] If \(\psi\in C^2(Q)\), then \(L_\psi\) is a Lagrangian submanifold of \((T^*Q,\omega)\). The key calculation is \[ \Phi_\psi^*\alpha=d\psi \quad\Longrightarrow\quad \Phi_\psi^*\omega=-d(\Phi_\psi^*\alpha)=-d(d\psi)=0, \] so \(\omega|_{L_\psi}\equiv 0\), while \(\dim L_\psi=n=\frac12\dim T^*Q\). In this sense, a Legendre graph is not merely a coordinate relation; it is a maximally isotropic geometric object in phase space [2512.19409]. This geometric usage is closely aligned with broader Legendre-submanifold language in contact and thermodynamic geometry. In Geometrothermodynamics, the equilibrium space \(\mathcal E\) is a maximal Legendre submanifold embedded by \(\varphi^*(\eta)=0\), reproducing \[ \Phi=\Phi(q^a),\qquad p_a=\frac{\partial \Phi}{\partial q^a}. \] This suggests that the cotangent-bundle Legendre graph is one exact-differential realization of a more general Legendre-submanifold paradigm [1404.6296]. ## 3. Preservation theorems and Legendre dynamics The geometric notion becomes dynamical in the definition of Legendre dynamics. If the system state at time \(k\) is represented by \((q_k,p_k)\), then the trajectory is Legendre-dynamic if there exists a potential \(\psi_k\) such that \[ (q_k,p_k)\in L_{\psi_k}\qquad\text{for all }k, \] equivalently, \[ p_k=d\psi_k(q_k). \] The invariant is therefore not the individual potential alone, but membership in the family of graphs of exact differentials. The paper distinguishes strong Legendre dynamics, where the same potential \(\psi\) persists, from the general case in which the potential may change to \(\psi_{k+1}\) after an update [2512.19409]. A central classification theorem states that a symplectomorphism \(F:T^*Q\to T^*Q\) preserves Legendre type structure if and only if \[ F=\tau_{d\chi}\circ f^\sharp, \] where \(f:Q\to Q\) is a diffeomorphism, \(f^\sharp\) is the cotangent lift \[ f^\sharp(q,p)=\bigl(f(q),(f^{-1})^*p\bigr), \] and \(\tau_{d\chi}\) is exact fiber translation. If \(L_\psi\) is the original graph, then \[ F(L_\psi)=L_{\psi'} \qquad\text{with}\qquad \psi'=\psi\circ f^{-1}+\chi. \] Hence the admissible maps are exactly those symplectomorphisms that transport one graph-of-a-gradient to another graph-of-a-gradient [2512.19409]. The same normal form appears in the paper’s Symplectic Reservoir construction. For Hamiltonians at most linear in momentum, \[ H(q,p)=\langle p,A(q)\rangle+V(q), \] the flow map satisfies \[ \phi_t=\tau_{d\chi_t}\circ f_t^\sharp. \] Therefore the reservoir update acts by \[ L_\psi \mapsto L_{\psi\circ f^{-1}+\chi}, \] preserving Legendre duality at each time step. The paper states that the associated class of Legendre dynamics includes linear time-invariant Gaussian process regression and Ornstein–Uhlenbeck dynamics, and its main theorem shows that every Symplectic Reservoir update has this normal form [2512.19409]. ## 4. Paley-type graphs governed by the Legendre symbol In arithmetic graph theory, the Legendre symbol provides another route to a “Legendre graph.” For an odd prime \(p\), the classical Paley graph \(P_q\) is defined on \(\mathbb{F}_q\), with \(u\) adjacent to \(v\) if \(u-v\) is a nonzero square in \(\mathbb{F}_q\). The generalized construction replaces the prime-field quadratic character by a primitive quadratic Dirichlet character \(\chi_\Delta\) of conductor \(D=|\Delta|\). The generalized Paley graph \(P_\Delta\) has \[ V(P_\Delta)=\{0,1,\dots,D-1\} \] and adjacency rule \[ (u,v)\text{ is an edge } \iff \chi_\Delta(v-u)=1. \] Because adjacency depends only on \(v-u\pmod D\), \(P_\Delta\) is a circulant graph on \(\mathbb{Z}/D\mathbb{Z}\). When \(\Delta>0\), \(\chi_\Delta(-1)=1\) and the graph is undirected; when \(\Delta<0\), \(\chi_\Delta(-1)=-1\) and it is directed [2212.02005]. In the prime-modulus special case, \(\chi_\Delta\) is the Legendre symbol. A natural interpretation is therefore that a Legendre graph in this setting is a Cayley graph whose edges are determined by whether differences are quadratic residues according to the Legendre symbol. The degree is \[ \deg(P_\Delta)=\frac12\varphi(D), \] and the adjacency spectrum is explicitly diagonalized by Fourier modes. For a \(D\)-th root of unity \(\omega\), the eigenvalue is \[ \lambda(\omega)=\frac12\sum_{a=0}^{D-1}\chi(a)\bigl(1+\chi(a)\bigr)\omega^a. \] Using quadratic Gauss sums \[ G(b,\chi)=\sum_{a=1}^{D-1}\chi(a)\zeta_D^{ab}, \qquad \zeta_D=e^{2\pi i/D}, \] the paper derives the explicit spectrum \[ \left[\frac12\frac{\varphi(D)}{\varphi(d)}\mu(d)\right]_{\varphi(d)} \quad (d\mid D,\ d