--- title: Edge-Convex Multi-Invariants url: https://www.emergentmind.com/topics/edge-convex-multi-invariants type: topic --- # Edge-Convex Multi-Invariants 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 [1012.0623] [2402.13893] [2509.06348] [2510.14272] [2605.28981]. ## 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 $J(e)=(\Delta_e d,\Delta_e\delta,\Delta_e\sigma)$ [1012.0623] [2402.13893] [2101.04008] [2605.28981]. 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 [1012.0623]. The basic elementary invariants are $$ \Theta_P(A)=\max_{\Pi\in\mathrm{Sym}(n)}\langle P,\Pi A\Pi^T\rangle, $$ with $\langle X,Y\rangle=\mathrm{tr}(XY)$. The central representation theorem states that every convex graph invariant can be written as $$ f(A)=\sup_{P\in\mathcal{P}}\big(\Theta_P(A)-\alpha_P\big), $$ for a suitable index set $\mathcal{P}\subseteq\mathbf{S}$ and scalars $\alpha_P$. The analogous statement for invariant convex sets gives $$ \mathcal{S}=\bigcap_{P\in\mathcal{P}}\{A\in\mathbf{S}:\Theta_P(A)\le \alpha_P\}. $$ This makes elementary invariants the support-function building blocks of the convex hull of the relabeling orbit of $A$ [1012.0623]. Several standard graph quantities fit this framework. The maximum degree $$ \Delta(A)=\max_i\sum_{j=1}^n A_{ij} $$ is convex and invariant. The MAXCUT value is a convex graph invariant in $A$, and its tractable semidefinite relaxation can be written, up to shift and rescaling, as $$ \max_{X\succeq 0,\ \mathrm{diag}(X)=\mathbf{1}} \frac14\sum_{i,j}A_{ij}(1-X_{ij}), $$ or equivalently in the convex-in-$A$ form $$ f(A)=\min_{X\in\mathbf{S}}\langle X,A\rangle\quad\text{s.t. }X_{ii}=1,\ X\succeq 0. $$ Spectral invariants also form a central subclass; for instance, $$ \sum_{i=1}^k \lambda_i(A)=\max_{U^TU=I_k}\mathrm{tr}(U^TAU) $$ 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 [1012.0623]. The “multi-invariant” aspect is explicit. If $g_1,\dots,g_m$ are convex graph invariants, then $$ g(A)=\sum_{i=1}^m \alpha_i g_i(A) $$ is again convex and invariant, and the intersection $$ \{A\in\mathbf{S}: g_i(A)\le c_i,\ i=1,\dots,m\} $$ is an invariant convex feasible set. This supports joint enforcement of degree bounds, cut values, spectral gaps, and forbidden-subgraph surrogates. Computationally, evaluating $\Theta_P(A)$ is a Quadratic Assignment Problem, but the paper gives a spectral relaxation $$ \Lambda_P(A)=\lambda(P)^T\lambda(A) $$ and an SDP relaxation $\Omega_P(A)\ge \Theta_P(A)$, with symmetry reduction from $O(n^2)$ to $O(kn)$ when $P$ is a small pattern. Applications include graph deconvolution, hypothesis testing between graph families, and graph generation; the deconvolution experiments reported $100/100$ successes for $16$-cycle + Clebsch graph, $96/100$ for $16$-cycle + Shrikhande, and $94/100$ for Clebsch + Shrikhande [1012.0623]. ## 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 $K\lambda$ of a semisimple connected compact Lie group $K$. For a subset $X$ of a Euclidean space and $r\in\mathbb{N}$, the $r$-th partial convex hull is $$ C_r(X)=\operatorname{Conv}\{x_1+\cdots+x_r:x_1,\ldots,x_r\in X\}. $$ Specializing to $X=K\lambda$, the paper defines two numerical invariants: $$ r_0(\lambda)=\min\{r\in\mathbb{N}:0\in C_r(K\lambda)\}, $$ and $$ r(\lambda)=\min\{r\in\mathbb{N}:\operatorname{Conv}(K\lambda)=C_r(K\lambda)\}. $$ Here the multi-invariant is the pair $(r_0(\lambda),r(\lambda))$, and the “edge” geometry is controlled by exposed faces and Littlewood–Richardson-type facet inequalities [2402.13893]. These invariants are governed by moment-map convexity. For the $r$-fold diagonal embedding, the Littlewood–Richardson cone is $$ CLR_r(K)=\operatorname{Conv}(LR_r)=\{(\lambda_1,\ldots,\lambda_r)\in(\mathfrak{t}_+)^{\times r}:0\in K\lambda_1+\cdots+K\lambda_r\}. $$ The characterization $$ r\ge r_0(\lambda)\iff (\lambda,\ldots,\lambda)\in CLR_r $$ implies that $$ A_r=\{\lambda\in\mathfrak{t}_+:r_0(\lambda)\le r\} $$ is a rational convex polyhedral cone. Likewise, $$ C_r=\{\lambda\in\mathfrak{t}_+:r(\lambda)\le r\} $$ is a rational convex polyhedral cone, and $r(\lambda)\le r$ is equivalent to convexity of the $r$-fold orbit sum and to a family of Levi-factor conditions [2402.13893]. The maximal-face formula $$ r(\lambda)=\max_{\Pi'\subset \Pi} r_0(\lambda|_{\Pi'}) $$ shows that the second invariant is computed by projecting to all Levi factors and taking the worst case. A Carathéodory-type bound gives $$ r(\lambda)\le \ell+1,\qquad C_{\ell+1}(K\lambda)=\operatorname{Conv}(K\lambda), $$ where $\ell=\dim\mathfrak{t}$. The paper also proves lower bounds for invariant-theoretic degrees: $$ r_0(\lambda)\le d_1(q\lambda),\qquad b_1(\lambda)\ge r_0(\lambda). $$ Concrete families were computed. If $w_0=-\mathrm{id}$, namely in types $B_\ell,C_\ell,D_{2m},E_7,E_8,F_4,G_2$, then $r_0(\lambda)=2$ for all $\lambda\neq 0$. For $SU(n)$, the formulas for fundamental weights include $$ r_0(\varpi_1)= \begin{cases} n,& n\text{ even},\\ n+2,& n\text{ odd}, \end{cases} \qquad r_0(\varpi_2)=2. $$ 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 [2402.13893]. ## 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 $$ I_m=J_1^m\cap\cdots\cap J_s^m $$ from a fixed decomposition $I=J_1\cap\cdots\cap J_s$, and proves the limit-shape theorem $$ C(\mathcal{I})=\bigcup_{m\ge 1}\frac1m NP(I_m)=NP(J_1)\cap\cdots\cap NP(J_s). $$ For symbolic powers this yields the symbolic polyhedron $SP(I)$, and for irreducible powers the irreducible polyhedron $IP(I)$. The Waldschmidt constant and the naive Waldschmidt constant are then linear programs: $$ \widehat{\alpha}(I)=\min\{y_1+\cdots+y_n:y\in SP(I)\}, $$ $$ \widetilde{\alpha}(I)=\min\{y_1+\cdots+y_n:y\in IP(I)\}, $$ with $$ \widetilde{\alpha}(I)\le \widehat{\alpha}(I)\le \alpha(I). $$ The same convex viewpoint naturally produces directional invariants $\alpha_c(\mathcal{I})=\min\{c\cdot y:y\in C(\mathcal{I})\}$ for $c\in\mathbb{R}^n_{\ge 0}$, making the multi-invariant interpretation explicit [2101.04008]. Graph ideals sharpen this polyhedral perspective. For a graph $G$, the binomial edge ideal is $$ J_G=(x_i y_j-x_j y_i\mid \{i,j\}\in E(G),\ i0}-\mathbf{1}_{m_i>0}\big)=a-r, $$ where $a$ is the number of support sizes newly introduced and $r$ the number that disappear, so $\Delta_e\sigma\in\{-2,-1,0,1,2\}$. The same paper proves a threshold-crossing principle for integer-valued invariants: $\Delta_eF\neq 0$ if and only if the edge crosses at least one threshold layer $L_F^{\ge r}(n)=\{\lambda:F(\lambda)\ge r\}$, and $|\Delta_eF|$ counts the number of integer thresholds crossed. For real-valued $F$, orienting edges toward increasing $F$ gives a strict gradient orientation, and every such orientation is acyclic [2605.28981]. A related but distinct edgewise strategy appears in the study of strongly regular graphs. For a vertex $a$, let $A|_{N_a}$ be the adjacency matrix restricted to the neighborhood $N_a$. The paper introduces $$ t_p(a)=\mathrm{tr}\big((A|_{N_a})^p\big),\qquad d_p(a)=\mathrm{sort}\big(\mathrm{diag}((A|_{N_a})^p)\big), $$ and also edge invariants built from the matrix $$ \bar A_{ab,cd}=A_{ab}A_{ac}A_{bd}, $$ with $$ T_p^{\mathrm{edge}}=\mathrm{tr}(\bar A^p),\qquad D_p^{\mathrm{edge}}=\mathrm{diag}(\bar A^p). $$ On Edward Spence’s dataset of $43{,}717$ strongly regular graphs, the vertex invariants alone distinguished all but $4$ pairs; with $p=3$ they fully distinguished $6$ of the $13$ parameter sets, and with $p=4$ this increased to $8$ of the $13$ sets. The remaining four pairs were resolved by the edge invariants at $p\in\{4,5\}$ [2402.04916]. 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 $\psi$ and $\bar\psi$ and represented by a bipartite $\psi$-graph with one edge color per party. For a connected $\psi$-graph $\mathcal{I}$, edge-convexity is defined by the existence, for each color $A$, of PSD matrices $M^{(k)}$ over reflecting cuts satisfying $$ \sum_{k\ \mathrm{s.t.}\ e\in R_k,\ e'\in L_k} M^{(k)}_{e,e'}=1 \qquad\forall\, e,e'. $$ If the graph is edge-convex for all colors, then $$ \hat{\nu}(\mathcal{I})=1-\hat{\mathcal{I}},\qquad \hat{\mathcal{I}}=\mathcal{I}^{1/n_r}, $$ 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 $A_n$, $B_n(=C_n)$, and $D_n$, and records six remaining cases: $E_6,E_7,E_8,F_4,H_3,H_5$ [2509.06348]. Variational convexity provides another manifestation. For discrete Laplace-type equations induced by integrable quad-equations on a white-diagonal graph, one has an action $$ S=\sum_{e=(x,y)}L(X,Y;\theta_e)-\sum_x \Theta_x X $$ with explicit Hessians depending on edge labels $\theta_e$. The paper derives sufficient, and often necessary, conditions on edge-label ranges under which $S$ is strictly convex or strictly concave, a key input for uniqueness and existence in Dirichlet problems. In the $(Q3)$ cases these functionals recover Euclidean and hyperbolic circle-pattern functionals, so edge-label convexity becomes a geometric existence theorem for circle patterns [1111.6273]. Several other literatures exhibit the same motif. In convex mosaics, the invariant is edge density, and the sharp minima are $3$ for translative unit-volume mosaics, attained by the face-to-face cube tiling, and $3\sqrt{3}/2$ for normal decomposable unit-volume mosaics, attained by face-to-face tilings with regular triangle-based right prisms [2312.08050]. In edge polytopes, $$ P(G)=\operatorname{conv}\{e_i+e_j:\{i,j\}\in E(G)\}, $$ graph-theoretic data determine convex-geometric invariants such as dimension, neighborliness, and facet count; for example, $P(G)$ is $k$-neighborly exactly when $G$ is $\mathcal{F}_k$-free, and there are edge polytopes with exponentially many facets [1307.6708]. In knot Floer theory, the Upsilon invariant is a piecewise linear convex function on $[0,2]$, and the paper constructs infinitely many mutually non-concordant hyperbolic knots $K_n^{(3,q)}$ whose $\Upsilon$ is convex because their $CFK^\infty$ complexes are stably equivalent to staircase complexes of $T(3,q)$ [2403.13342]. 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. Source: https://www.emergentmind.com/topics/edge-convex-multi-invariants