Edge-Convex Multi-Invariants

• Edge-convex multi-invariants are a family of constructions that aggregate edge data into convex bodies to jointly analyze several scalar graph invariants.
• They leverage tools such as convex optimization, polyhedral asymptotics, and moment-map convexity to handle invariance under relabeling and symmetry constraints.
• This framework unifies diverse applications from graph deconvolution and spectral analysis to representation theory and algebraic invariants in combinatorial settings.

Edge-convex multi-invariants are best understood as a family of constructions rather than a single standardized object. Across the literature, the expression is used for frameworks in which several invariants are organized through convexity and edge-level structure: convex functions of adjacency matrices invariant under relabeling, numerical invariants derived from partial convex hulls of orbits, polyhedral asymptotic invariants of graph ideals, edgewise jump signatures on combinatorial state graphs, and graph-labeled polynomial invariants whose edge-convexity implies monotonicity under local operations. This suggests a unifying theme: local edge data are aggregated into a convex body, convex cone, or convex optimization problem, and multiple scalar invariants are then studied jointly rather than in isolation (Chandrasekaran et al., 2010, Tsanov, 2024, Gadde et al., 8 Sep 2025, Belotserkovskiy et al., 16 Oct 2025, Lyudogovskiy, 27 May 2026).

1. Conceptual scope and recurring structure

Two recurring patterns are visible in the literature. In the first, an “edge-convex” invariant is literally convex in edge variables, such as the adjacency or edge-weight matrix, while invariance means equivariance under relabeling, group action, or another symmetry. In the second, “edge-convex” refers to constraints on how several invariants change across edges of a graph or graph-like object, often encoded by a convex cone of admissible jump vectors or by positivity conditions attached to reflecting cuts.

A common template, suggested by these works, consists of four ingredients. First, one fixes a structured object: a graph, coadjoint orbit, symbolic polyhedron, partition graph, or $\psi$-graph. Second, one defines several scalar invariants, such as degree, support size, spectral quantities, or asymptotic slopes. Third, one organizes these invariants by a convex object—support functions, polyhedral cones, Newton or symbolic polyhedra, or PSD feasibility conditions. Fourth, one studies weighted sums, intersections of sublevel sets, threshold crossings, or gradient orientations induced by these invariants. The resulting “multi-invariant” viewpoint is explicitly present in convex graph invariants, in the pair $(r_0(\lambda),r(\lambda))$ attached to coadjoint orbits, in symbolic-polyhedral asymptotics, and in jump signatures such as $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$ (Chandrasekaran et al., 2010, Tsanov, 2024, Camarneiro et al., 2021, Lyudogovskiy, 27 May 2026).

A plausible implication is that the term is best treated encyclopedically as a cross-domain methodology. The surveyed literature does not supply a single universal formal definition valid in all settings; instead, it supplies several precise domain-specific definitions linked by convexity, edge structure, and simultaneous treatment of multiple invariants.

2. Convex graph invariants on adjacency matrices

In graph optimization, the foundational definition is the convex graph invariant. Let $\mathbf{S}$ denote the set of real symmetric $n\times n$ matrices, and represent an undirected weighted graph by an adjacency matrix $A\in\mathbf{S}$. A function $f:\mathbf{S}\to\mathbb{R}$ is a convex graph invariant if it is convex and satisfies

$f(\Pi A\Pi^T)=f(A)$

for every permutation matrix $\Pi\in\mathrm{Sym}(n)$. In this sense, “edge-convex” means convexity in adjacency or edge weights together with permutation invariance (Chandrasekaran et al., 2010).

The basic elementary invariants are

$$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$

with $(r_0(\lambda),r(\lambda))$0. The central representation theorem states that every convex graph invariant can be written as

$(r_0(\lambda),r(\lambda))$1

for a suitable index set $(r_0(\lambda),r(\lambda))$2 and scalars $(r_0(\lambda),r(\lambda))$3. The analogous statement for invariant convex sets gives

$(r_0(\lambda),r(\lambda))$4

This makes elementary invariants the support-function building blocks of the convex hull of the relabeling orbit of $(r_0(\lambda),r(\lambda))$5 (Chandrasekaran et al., 2010).

Several standard graph quantities fit this framework. The maximum degree

$(r_0(\lambda),r(\lambda))$6

is convex and invariant. The MAXCUT value is a convex graph invariant in $(r_0(\lambda),r(\lambda))$7, and its tractable semidefinite relaxation can be written, up to shift and rescaling, as

$(r_0(\lambda),r(\lambda))$8

or equivalently in the convex-in-$(r_0(\lambda),r(\lambda))$9 form

$J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$0

Spectral invariants also form a central subclass; for instance,

$J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$1

is convex and invariant. The paper also highlights the number of edges, degree-sequence monotone linear functionals, the Laplacian spectral gap as a concave invariant whose superlevel sets are invariant convex sets, and the inverse stability number via the Motzkin–Straus formula and an SDP relaxation (Chandrasekaran et al., 2010).

The “multi-invariant” aspect is explicit. If $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$2 are convex graph invariants, then

$J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$3

is again convex and invariant, and the intersection

$J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$4

is an invariant convex feasible set. This supports joint enforcement of degree bounds, cut values, spectral gaps, and forbidden-subgraph surrogates. Computationally, evaluating $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$5 is a Quadratic Assignment Problem, but the paper gives a spectral relaxation

$J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$6

and an SDP relaxation $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$7, with symmetry reduction from $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$8 to $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$9 when $\mathbf{S}$0 is a small pattern. Applications include graph deconvolution, hypothesis testing between graph families, and graph generation; the deconvolution experiments reported $\mathbf{S}$1 successes for $\mathbf{S}$2-cycle + Clebsch graph, $\mathbf{S}$3 for $\mathbf{S}$4-cycle + Shrikhande, and $\mathbf{S}$5 for Clebsch + Shrikhande (Chandrasekaran et al., 2010).

3. Partial convex hulls and orbit-based multi-invariants

A second precise use of the term appears in representation theory, where the relevant object is a coadjoint orbit $\mathbf{S}$6 of a semisimple connected compact Lie group $\mathbf{S}$7. For a subset $\mathbf{S}$8 of a Euclidean space and $\mathbf{S}$9, the $n\times n$0-th partial convex hull is

$n\times n$1

Specializing to $n\times n$2, the paper defines two numerical invariants:

$n\times n$3

and

$n\times n$4

Here the multi-invariant is the pair $n\times n$5, and the “edge” geometry is controlled by exposed faces and Littlewood–Richardson-type facet inequalities (Tsanov, 2024).

These invariants are governed by moment-map convexity. For the $n\times n$6-fold diagonal embedding, the Littlewood–Richardson cone is

$n\times n$7

The characterization

$n\times n$8

implies that

$n\times n$9

is a rational convex polyhedral cone. Likewise,

$A\in\mathbf{S}$0

is a rational convex polyhedral cone, and $A\in\mathbf{S}$1 is equivalent to convexity of the $A\in\mathbf{S}$2-fold orbit sum and to a family of Levi-factor conditions (Tsanov, 2024).

The maximal-face formula

$A\in\mathbf{S}$3

shows that the second invariant is computed by projecting to all Levi factors and taking the worst case. A Carathéodory-type bound gives

$A\in\mathbf{S}$4

where $A\in\mathbf{S}$5. The paper also proves lower bounds for invariant-theoretic degrees:

$A\in\mathbf{S}$6

Concrete families were computed. If $A\in\mathbf{S}$7, namely in types $A\in\mathbf{S}$8, then $A\in\mathbf{S}$9 for all $f:\mathbf{S}\to\mathbb{R}$0. For $f:\mathbf{S}\to\mathbb{R}$1, the formulas for fundamental weights include

$f:\mathbf{S}\to\mathbb{R}$2

This framework therefore treats “multi-invariants” literally as a convex-geometric pair of orbit invariants whose values and feasible regions are encoded by rational polyhedral cones (Tsanov, 2024).

4. Polyhedral asymptotics for graph and monomial ideals

A third major realization of edge-convex multi-invariants is polyhedral. For a monomial ideal, the paper on asymptotic invariants for powers from decompositions defines graded families

$f:\mathbf{S}\to\mathbb{R}$3

from a fixed decomposition $f:\mathbf{S}\to\mathbb{R}$4, and proves the limit-shape theorem

$f:\mathbf{S}\to\mathbb{R}$5

For symbolic powers this yields the symbolic polyhedron $f:\mathbf{S}\to\mathbb{R}$6, and for irreducible powers the irreducible polyhedron $f:\mathbf{S}\to\mathbb{R}$7. The Waldschmidt constant and the naive Waldschmidt constant are then linear programs:

$f:\mathbf{S}\to\mathbb{R}$8

$f:\mathbf{S}\to\mathbb{R}$9

with

$f(\Pi A\Pi^T)=f(A)$0

The same convex viewpoint naturally produces directional invariants $f(\Pi A\Pi^T)=f(A)$1 for $f(\Pi A\Pi^T)=f(A)$2, making the multi-invariant interpretation explicit (Camarneiro et al., 2021).

Graph ideals sharpen this polyhedral perspective. For a graph $f(\Pi A\Pi^T)=f(A)$3, the binomial edge ideal is

$f(\Pi A\Pi^T)=f(A)$4

The paper studies $f(\Pi A\Pi^T)=f(A)$5, its lexicographic initial ideal $f(\Pi A\Pi^T)=f(A)$6, and its multigraded generic initial ideal $f(\Pi A\Pi^T)=f(A)$7 through symbolic polyhedra. For squarefree monomial ideals $f(\Pi A\Pi^T)=f(A)$8, the symbolic polyhedron satisfies

$f(\Pi A\Pi^T)=f(A)$9

and the asymptotic invariants are read off from vertices:

$\Pi\in\mathrm{Sym}(n)$0

For binomial edge ideals, the main universal result is

$\Pi\in\mathrm{Sym}(n)$1

Moreover,

$\Pi\in\mathrm{Sym}(n)$2

where $\Pi\in\mathrm{Sym}(n)$3 is the edge ideal, and for non-empty $\Pi\in\mathrm{Sym}(n)$4 one has

$\Pi\in\mathrm{Sym}(n)$5

For complete graphs, $\Pi\in\mathrm{Sym}(n)$6; for complete $\Pi\in\mathrm{Sym}(n)$7-partite graphs, $\Pi\in\mathrm{Sym}(n)$8; and for bipartite graphs this gives $\Pi\in\mathrm{Sym}(n)$9. The paper also proves that vertices of $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$0 and $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$1 decompose through connected induced subgraphs, so global asymptotic invariants are extremized over local edge-based convex sets (Belotserkovskiy et al., 16 Oct 2025).

This suggests that one robust meaning of “edge-convex multi-invariants” is polyhedral asymptotics in which edge-generated algebraic objects are controlled by convex bodies whose vertices or facets simultaneously encode several invariants.

5. Edgewise jump signatures and local combinatorial discrimination

A different, explicitly edgewise, meaning appears when invariants are not merely evaluated on objects but differentiated along edges of a state graph. In the partition graph $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$2, whose vertices are partitions of $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$3 and whose edges are elementary transfers of one unit between parts, the paper defines for any vertex invariant $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$4 the signed jump

$$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$5

on an oriented edge $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$6. For the degree $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$7, local simplex dimension $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$8, and support size $$\Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle,$$9, the basic jump signature is

$(r_0(\lambda),r(\lambda))$00

The absolute signature $(r_0(\lambda),r(\lambda))$01 is an unoriented invariant, and the transition rank $(r_0(\lambda),r(\lambda))$02 counts how many components are nonzero (Lyudogovskiy, 27 May 2026).

The sharp universal theorem concerns support: for every edge,

$(r_0(\lambda),r(\lambda))$03

More precisely,

$(r_0(\lambda),r(\lambda))$04

where $(r_0(\lambda),r(\lambda))$05 is the number of support sizes newly introduced and $(r_0(\lambda),r(\lambda))$06 the number that disappear, so $(r_0(\lambda),r(\lambda))$07. The same paper proves a threshold-crossing principle for integer-valued invariants: $(r_0(\lambda),r(\lambda))$08 if and only if the edge crosses at least one threshold layer $(r_0(\lambda),r(\lambda))$09, and $(r_0(\lambda),r(\lambda))$10 counts the number of integer thresholds crossed. For real-valued $(r_0(\lambda),r(\lambda))$11, orienting edges toward increasing $(r_0(\lambda),r(\lambda))$12 gives a strict gradient orientation, and every such orientation is acyclic (Lyudogovskiy, 27 May 2026).

A related but distinct edgewise strategy appears in the study of strongly regular graphs. For a vertex $(r_0(\lambda),r(\lambda))$13, let $(r_0(\lambda),r(\lambda))$14 be the adjacency matrix restricted to the neighborhood $(r_0(\lambda),r(\lambda))$15. The paper introduces

$(r_0(\lambda),r(\lambda))$16

and also edge invariants built from the matrix

$(r_0(\lambda),r(\lambda))$17

with

$(r_0(\lambda),r(\lambda))$18

On Edward Spence’s dataset of $(r_0(\lambda),r(\lambda))$19 strongly regular graphs, the vertex invariants alone distinguished all but $(r_0(\lambda),r(\lambda))$20 pairs; with $(r_0(\lambda),r(\lambda))$21 they fully distinguished $(r_0(\lambda),r(\lambda))$22 of the $(r_0(\lambda),r(\lambda))$23 parameter sets, and with $(r_0(\lambda),r(\lambda))$24 this increased to $(r_0(\lambda),r(\lambda))$25 of the $(r_0(\lambda),r(\lambda))$26 sets. The remaining four pairs were resolved by the edge invariants at $(r_0(\lambda),r(\lambda))$27 (Duda, 2024).

The surveyed literature therefore supports a second major reading of the topic: an edge-convex multi-invariant may be a vector of edgewise changes or local edge-lifted signatures, often organized by cones of allowed jumps, by threshold layers, or by convex combinations of several scalar features.

6. Monotonicity, variational convexity, and broader manifestations

In multipartite quantum information, the term becomes fully formalized. A multi-invariant is a local unitary invariant polynomial obtained from copies of $(r_0(\lambda),r(\lambda))$28 and $(r_0(\lambda),r(\lambda))$29 and represented by a bipartite $(r_0(\lambda),r(\lambda))$30-graph with one edge color per party. For a connected $(r_0(\lambda),r(\lambda))$31-graph $(r_0(\lambda),r(\lambda))$32, edge-convexity is defined by the existence, for each color $(r_0(\lambda),r(\lambda))$33, of PSD matrices $(r_0(\lambda),r(\lambda))$34 over reflecting cuts satisfying

$(r_0(\lambda),r(\lambda))$35

If the graph is edge-convex for all colors, then

$(r_0(\lambda),r(\lambda))$36

is a pure-state entanglement monotone on average under LOCC. The paper conjectures that edge-convex multi-invariants are precisely those labeled by finite Coxeter groups, proves this for $(r_0(\lambda),r(\lambda))$37, $(r_0(\lambda),r(\lambda))$38, and $(r_0(\lambda),r(\lambda))$39, and records six remaining cases: $(r_0(\lambda),r(\lambda))$40 (Gadde et al., 8 Sep 2025).

Variational convexity provides another manifestation. For discrete Laplace-type equations induced by integrable quad-equations on a white-diagonal graph, one has an action

$(r_0(\lambda),r(\lambda))$41

with explicit Hessians depending on edge labels $(r_0(\lambda),r(\lambda))$42. The paper derives sufficient, and often necessary, conditions on edge-label ranges under which $(r_0(\lambda),r(\lambda))$43 is strictly convex or strictly concave, a key input for uniqueness and existence in Dirichlet problems. In the $(r_0(\lambda),r(\lambda))$44 cases these functionals recover Euclidean and hyperbolic circle-pattern functionals, so edge-label convexity becomes a geometric existence theorem for circle patterns (Bobenko et al., 2011).

Several other literatures exhibit the same motif. In convex mosaics, the invariant is edge density, and the sharp minima are $(r_0(\lambda),r(\lambda))$45 for translative unit-volume mosaics, attained by the face-to-face cube tiling, and $(r_0(\lambda),r(\lambda))$46 for normal decomposable unit-volume mosaics, attained by face-to-face tilings with regular triangle-based right prisms (Kadlicskó et al., 2023). In edge polytopes,

$(r_0(\lambda),r(\lambda))$47

graph-theoretic data determine convex-geometric invariants such as dimension, neighborliness, and facet count; for example, $(r_0(\lambda),r(\lambda))$48 is $(r_0(\lambda),r(\lambda))$49-neighborly exactly when $(r_0(\lambda),r(\lambda))$50 is $(r_0(\lambda),r(\lambda))$51-free, and there are edge polytopes with exponentially many facets (Tran et al., 2013). In knot Floer theory, the Upsilon invariant is a piecewise linear convex function on $(r_0(\lambda),r(\lambda))$52, and the paper constructs infinitely many mutually non-concordant hyperbolic knots $(r_0(\lambda),r(\lambda))$53 whose $(r_0(\lambda),r(\lambda))$54 is convex because their $(r_0(\lambda),r(\lambda))$55 complexes are stably equivalent to staircase complexes of $(r_0(\lambda),r(\lambda))$56 (Himeno, 2024).

The surveyed literature therefore suggests that “edge-convex multi-invariants” names a broad research pattern: convexity is imposed at the level of edges, edge labels, edge weights, or edge transitions, and multiple invariants are then coupled through support functions, polyhedral cones, PSD feasibility, or gradient structures. A common misconception would be to treat the phrase as a single canonical invariant. The evidence points instead to a family of highly structured constructions whose unity lies in method rather than in a universal formula.