Legendre Graph: Geometry & Quadratic Symbols

• Legendre graph is a structured object defined in symplectic geometry as the graph of the differential of a convex potential, forming a Lagrangian submanifold in the cotangent bundle.
• In arithmetic graph theory, Legendre graphs are Paley-type circulant graphs where adjacency is determined by quadratic residue conditions via the Legendre symbol or quadratic Dirichlet characters.
• In graph labeling, Legendre graphs serve as auxiliary constructions that guide cordial labeling rules by applying the Legendre symbol to sums of vertex labels.

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 (Fong et al., 22 Dec 2025, Mináč et al., 2022, Andoyo, 11 Sep 2025).

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 (Fong et al., 22 Dec 2025, Mináč et al., 2022, Andoyo, 11 Sep 2025).

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_n^k(f,p)$0

In local coordinates $L_n^k(f,p)$1, this is

$L_n^k(f,p)$2

The same construction is presented as a duality of Legendre type: $L_n^k(f,p)$3 are primal coordinates, $L_n^k(f,p)$4 are dual coordinates, and the duality relation is $L_n^k(f,p)$5. In the information-geometric formulation, the parameter manifold is denoted by $L_n^k(f,p)$6, a dualistic model $L_n^k(f,p)$7 satisfies $L_n^k(f,p)$8, and the corresponding Legendre graph is

$L_n^k(f,p)$9

Thus the graph encodes the statement that dual coordinates are exact derivatives of a potential rather than independent variables (Fong et al., 22 Dec 2025).

The cotangent bundle $L_\psi\subset T^*Q$0 carries the canonical $L_\psi\subset T^*Q$1-form

$L_\psi\subset T^*Q$2

and the canonical symplectic form

$L_\psi\subset T^*Q$3

If $L_\psi\subset T^*Q$4, then $L_\psi\subset T^*Q$5 is a Lagrangian submanifold of $L_\psi\subset T^*Q$6. The key calculation is

$L_\psi\subset T^*Q$7

so $L_\psi\subset T^*Q$8, while $L_\psi\subset T^*Q$9. In this sense, a Legendre graph is not merely a coordinate relation; it is a maximally isotropic geometric object in phase space (Fong et al., 22 Dec 2025).

This geometric usage is closely aligned with broader Legendre-submanifold language in contact and thermodynamic geometry. In Geometrothermodynamics, the equilibrium space $p=d\psi(q)$0 is a maximal Legendre submanifold embedded by $p=d\psi(q)$1, reproducing

$p=d\psi(q)$2

This suggests that the cotangent-bundle Legendre graph is one exact-differential realization of a more general Legendre-submanifold paradigm (Garcia-Pelaez et al., 2014).

3. Preservation theorems and Legendre dynamics

The geometric notion becomes dynamical in the definition of Legendre dynamics. If the system state at time $p=d\psi(q)$3 is represented by $p=d\psi(q)$4, then the trajectory is Legendre-dynamic if there exists a potential $p=d\psi(q)$5 such that

$p=d\psi(q)$6

equivalently,

$p=d\psi(q)$7

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 $p=d\psi(q)$8 persists, from the general case in which the potential may change to $p=d\psi(q)$9 after an update (Fong et al., 22 Dec 2025).

A central classification theorem states that a symplectomorphism $\chi_\Delta(v-u)=1$0 preserves Legendre type structure if and only if

$\chi_\Delta(v-u)=1$1

where $\chi_\Delta(v-u)=1$2 is a diffeomorphism, $\chi_\Delta(v-u)=1$3 is the cotangent lift

$\chi_\Delta(v-u)=1$4

and $\chi_\Delta(v-u)=1$5 is exact fiber translation. If $\chi_\Delta(v-u)=1$6 is the original graph, then

$\chi_\Delta(v-u)=1$7

Hence the admissible maps are exactly those symplectomorphisms that transport one graph-of-a-gradient to another graph-of-a-gradient (Fong et al., 22 Dec 2025).

The same normal form appears in the paper’s Symplectic Reservoir construction. For Hamiltonians at most linear in momentum,

$\chi_\Delta(v-u)=1$8

the flow map satisfies

$\chi_\Delta(v-u)=1$9

Therefore the reservoir update acts by

$L_n^k(f,p)$0

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 (Fong et al., 22 Dec 2025).

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 $L_n^k(f,p)$1, the classical Paley graph $L_n^k(f,p)$2 is defined on $L_n^k(f,p)$3, with $L_n^k(f,p)$4 adjacent to $L_n^k(f,p)$5 if $L_n^k(f,p)$6 is a nonzero square in $L_n^k(f,p)$7. The generalized construction replaces the prime-field quadratic character by a primitive quadratic Dirichlet character $L_n^k(f,p)$8 of conductor $L_n^k(f,p)$9. The generalized Paley graph $\left(\frac{f(a)+f(b)}{p}\right)=k$0 has

$\left(\frac{f(a)+f(b)}{p}\right)=k$1

and adjacency rule

$\left(\frac{f(a)+f(b)}{p}\right)=k$2

Because adjacency depends only on $\left(\frac{f(a)+f(b)}{p}\right)=k$3, $\left(\frac{f(a)+f(b)}{p}\right)=k$4 is a circulant graph on $\left(\frac{f(a)+f(b)}{p}\right)=k$5. When $\left(\frac{f(a)+f(b)}{p}\right)=k$6, $\left(\frac{f(a)+f(b)}{p}\right)=k$7 and the graph is undirected; when $\left(\frac{f(a)+f(b)}{p}\right)=k$8, $\left(\frac{f(a)+f(b)}{p}\right)=k$9 and it is directed (Mináč et al., 2022).

In the prime-modulus special case, $n$0 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

$n$1

and the adjacency spectrum is explicitly diagonalized by Fourier modes. For a $n$2-th root of unity $n$3, the eigenvalue is

$n$4

Using quadratic Gauss sums

$n$5

the paper derives the explicit spectrum

$n$6

together with

$n$7

This arithmetic control permits a Ramanujan classification and an effective upper bound for the Cheeger number involving $n$8 (Mináč et al., 2022).

5. Legendre graphs in cordial labeling theory

A third usage appears in the study of Legendre cordial labeling modulo an odd prime $n$9. For a simple connected graph $Q$0 of order $Q$1, a bijection

$Q$2

induces an edge labeling

$Q$3

The labeling is Legendre cordial if

$Q$4

The corresponding Legendre graph is the auxiliary graph

$Q$5

defined for $Q$6 by

$Q$7

Thus $Q$8 records edges whose label sums are quadratic residues, while $Q$9 records those whose sums are quadratic nonresidues (Andoyo, 11 Sep 2025).

For complete graphs $\psi:Q\to\mathbb{R}$0, this decomposition is exact: edges in $\psi:Q\to\mathbb{R}$1 receive label $\psi:Q\to\mathbb{R}$2, edges in $\psi:Q\to\mathbb{R}$3 receive label $\psi:Q\to\mathbb{R}$4, and edges with endpoint sum congruent to $\psi:Q\to\mathbb{R}$5 also receive label $\psi:Q\to\mathbb{R}$6. The paper proves detailed degree formulas in $\psi:Q\to\mathbb{R}$7 after decomposing $\psi:Q\to\mathbb{R}$8 into blocks of size $\psi:Q\to\mathbb{R}$9. It also derives the size formula

$L_n^k(f,p)$00

where

$L_n^k(f,p)$01

and $L_n^k(f,p)$02 are explicit sums weighted by the Legendre symbol (Andoyo, 11 Sep 2025).

The main characterization is that $L_n^k(f,p)$03 is a Legendre cordial graph modulo $L_n^k(f,p)$04 if and only if, with $L_n^k(f,p)$05,

$L_n^k(f,p)$06

or

$L_n^k(f,p)$07

A strong special case is

$L_n^k(f,p)$08

The paper also defines

$L_n^k(f,p)$09

and reports that computational plots suggest $L_n^k(f,p)$10 tends to decrease as $L_n^k(f,p)$11 grows, motivating the conjecture

$L_n^k(f,p)$12

Subsequent work investigated Legendre cordial labeling for graphs produced by join, corona, lexicographic product, cartesian product, tensor product, and strong product, extending the labeling program in which the auxiliary Legendre graph operates (Andoyo, 13 Sep 2025).

6. Conceptual relations, distinctions, and recurrent themes

Across these usages, the term “Legendre graph” always encodes a structured relation rather than an arbitrary adjacency pattern. In the symplectic setting, the relation is exact duality,

$L_n^k(f,p)$13

and the object is a Lagrangian submanifold. In the generalized Paley setting, the relation is arithmetic and difference-based,

$L_n^k(f,p)$14

producing a circulant Cayley graph. In the cordial-labeling setting, the relation is sum-based,

$L_n^k(f,p)$15

and the graph is auxiliary to a labeling problem rather than intrinsic to the underlying graph (Fong et al., 22 Dec 2025, Mináč et al., 2022, Andoyo, 11 Sep 2025).

The terminological overlap is therefore substantive but limited. The geometric Legendre graph is not a graph-theoretic graph, and the arithmetic Legendre graphs do not arise from Legendre duality. Conversely, the graph-labeling construction depends on a chosen bijection $L_n^k(f,p)$16, so it is not a canonical graph family in the way generalized Paley graphs are. This suggests that the unifying role of the term is historical and symbolic: it signals either Legendre dual structure or Legendre-symbol arithmetic, but not a single universal definition.

A plausible implication is that future uses of the term will continue to bifurcate along these lines unless a discipline-specific qualifier is attached, such as “Legendre graph in $L_n^k(f,p)$17,” “Legendre-symbol graph,” or “Legendre graph $L_n^k(f,p)$18.” The existing literature already points in that direction by embedding each notion in a highly developed local theory—symplectic Legendre dynamics, quadratic-character graph theory, and Legendre cordial labeling—rather than in a unified cross-disciplinary definition.