--- title: Haemers Conjecture in Spectral Graph Theory url: https://www.emergentmind.com/topics/haemers-conjecture type: topic --- # Haemers Conjecture in Spectral Graph Theory “Haemers Conjecture” is not a single universally fixed statement. In recent graph-theoretic and spectral literature, the label is used for several distinct conjectures associated with Willem Haemers. The principal usages are: the Seidel energy conjecture, asserting that every graph \(G\) of order \(n\) satisfies \(E_S(G)\ge 2n-2\); the Laplacian toughness conjecture, asserting that every connected graph \(\Gamma\) satisfies \(t(\Gamma)\ge \mu_2/(\mu_n-\delta)\); and the van Dam–Haemers conjecture, asserting that almost all graphs are determined by their adjacency spectrum. The first two are now theorems, whereas the third remains open [1901.06692] [2605.15738] [2309.09788]. ## 1. Terminological scope and principal meanings The ambiguity of the expression reflects the breadth of Haemers’ influence across spectral graph theory, Seidel switching, zero-error information theory, and algebraic combinatorics. In one line of work, the conjecture concerns extremal Seidel energy. In another, it concerns a spectral lower bound for graph toughness. In a third, it concerns asymptotic spectral rigidity, namely whether almost all graphs are determined by spectrum. | Usage | Core statement | Status | |---|---|---| | **Seidel energy conjecture** | \(E_S(G)\ge 2n-2\) for every \(n\)-vertex graph | Proved for graphs; hypergraph analogue fails in general [1901.06692] [2606.17817] | | **Laplacian toughness conjecture** | \(t(\Gamma)\ge \mu_2/(\mu_n-\delta)\) for connected graphs | Proved [2605.15738] | | **van Dam–Haemers conjecture** | Almost all graphs are determined by their adjacency spectrum | Open; exponential lower bounds known [2309.09788] | The surrounding literature also contains “Haemers bounds” on Shannon capacity and several Haemers-style spectral inequalities that are not conjectures in the strict sense. This suggests that the phrase is intrinsically context-dependent. ## 2. Seidel energy conjecture For a simple graph \(G\) on \(n\) vertices, the Seidel matrix is \[ S(G)=J-I-2A(G), \] where \(A(G)\) is the adjacency matrix, \(J\) is the all-ones matrix, and \(I\) is the identity. If \(\theta_1(G),\dots,\theta_n(G)\) are the eigenvalues of \(S(G)\), then the Seidel energy is \[ \mathcal{E}_S(G)=\sum_{i=1}^n |\theta_i(G)|. \] Haemers’ conjecture in this setting states that for every graph \(G\) on \(n\) vertices, \[ \mathcal{E}_S(G)\ge 2n-2, \] with equality at the complete graph \(K_n\). Since \(S(K_n)=I-J\), its Seidel spectrum is \(\{1-n,1^{(n-1)}\}\), and therefore \(\mathcal{E}_S(K_n)=2n-2\) [1301.0075]. The conjecture was resolved in full generality: for every graph \(G\) of order \(n\), \[ E(S(G))\ge E(S(K_n))=2n-2, \] and if \(G\) and \(K_n\) are not SC-equivalent, then the inequality is strict. Here SC-equivalence means equivalence under Seidel switching, possibly followed by complementation; both operations preserve Seidel energy because switching gives \(S(G')=DS(G)D\) with diagonal \(D\), and complementation gives \(S(\overline G)=-S(G)\) [1901.06692]. Before the full proof, several partial results were known. Haemers proved the conjecture for \(n\le 10\), Greaves–Koolen–Munemasa–Szöllősi extended this to \(n\le 12\), Oboudi treated \(r\)-regular graphs under spectral restrictions, and Ghorbani proved that for every \(0<\alpha<2\), \[ \sum_{i=1}^n |\theta_i(G)|^{\alpha} \ge (n-1)^{\alpha}+(n-1) \quad\text{if and only if}\quad |\det S(G)|\ge n-1. \] At \(\alpha=1\), this yields the Seidel energy bound for all graphs satisfying \(|\det S(G)|\ge n-1\) [1301.0075] [2606.17817]. The full proof proceeds through an integral representation of \(p\)-energy, Cauchy–Binet lower bounds on elementary symmetric polynomials of \(S(G)^2\), and a combinatorial analysis of “odd pairs.” A key structural reduction uses Seidel switching to normalize the graph and force the existence of large minors unless the graph is SC-equivalent to \(K_n\) [1901.06692]. A notable extension is negative rather than positive. For \(k\)-uniform hypergraphs, the natural analogue of the Seidel matrix uses pair co-degrees, and the complete \(k\)-uniform hypergraph does not in general minimize Seidel energy. The paper constructs the \(k\)-uniform hypertriangle \(\mathcal{T}_3^k\) on \(n=3k-3\) vertices and shows \[ E_S(\mathcal{T}_3^k)1\Bigr\}, \] with the convention \(t(K_n)=\infty\). If \(L=D-A\) is the Laplacian matrix, with Laplacian eigenvalues \[ 0=\mu_1<\mu_2\le \cdots \le \mu_n, \] and \(\delta\) is the minimum degree, then Haemers conjectured that \[ t(\Gamma)\ge \frac{\mu_2}{\mu_n-\delta}. \] This conjecture has now been proved [2605.15738]. The theorem subsumes earlier spectral bounds, including those of Alon and Brouwer for regular graphs. In the \(d\)-regular case, since \(\mu_i=d-\lambda_i(A)\), one obtains \[ t(\Gamma)\ge \frac{d-\lambda_2(A)}{-\lambda_n(A)}, \] which is Brouwer’s conjectured bound, now recovered as a consequence of the Laplacian formulation [2605.15738]. The proof is based on a cut decomposition. For a vertex cut \(U\subseteq V\), let \(H_1,\dots,H_c\) be the components of \(\Gamma-U\), and define a quotient matrix for the partition \(\{H_1,\dots,H_c,U\}\). Two positive semidefinite matrices are central: \[ \mu_n I-Q_\pi(L(\Gamma)) \quad\text{and}\quad Q_\pi\Bigl(L(\Gamma)+\frac{\mu_2}{n}J\Bigr)-\mu_2 I. \] A Schur-complement argument yields \[ \sum_{i=1}^{c}\frac{e_i}{|H_i|}\le (\mu_n-\delta)|U|, \] where \(e_i\) is the number of edges from \(H_i\) to \(U\). A second PSD argument shows that at most one component can satisfy \(e_i/|H_i|<\mu_2\). The proof then splits into two cases and derives the contradiction needed to force \[ |U|(\mu_n-\delta)\ge c\,\mu_2 \] for every cut with \(c=c(\Gamma-U)\ge 2\), which implies the claimed toughness bound [2605.15738]. The bound is sharp for complete multipartite graphs. For \(K_{n_1,\dots,n_s}\) with \(n_1\ge\cdots\ge n_s\), \[ \mu_n=n,\qquad \mu_2=n-n_1,\qquad \delta=n-n_1, \] hence \[ \frac{\mu_2}{\mu_n-\delta}=\frac{n-n_1}{n_1}=t(K_{n_1,\dots,n_s}). \] The same paper also exhibits non-complete multipartite equality cases obtained by adding edges inside suitable parts while preserving \(\delta\), \(\mu_2\), and \(\mu_n\). A complete characterization of equality remains open [2605.15738]. The theorem is strongest when \(\mu_2\) is large and \(\mu_n-\delta\) is small. It can be loose on sparse low-connectivity graphs; for the cycle \(C_n\), the bound gives \[ t(C_n)\ge \frac{2-2\cos(2\pi/n)}{2}\approx \frac{2\pi^2}{n^2}, \] whereas \(t(C_n)=1\) [2605.15738]. ## 4. van Dam–Haemers conjecture on spectral determination A graph \(G\) is determined by its adjacency spectrum, or DS, if every graph \(H\) with the same adjacency spectrum is isomorphic to \(G\). The van Dam–Haemers conjecture asserts that almost all graphs are DS: if \(U_n\) denotes the set of unlabelled \(n\)-vertex graphs, then \[ \frac{|\{G\in U_n:\;G\text{ is DS}\}|}{|U_n|}\to 1 \qquad\text{as }n\to\infty. \] This is the asymptotic “almost all graphs are determined by their spectrum” conjecture [2309.09788]. The strongest general quantitative progress in the supplied literature gives an exponential lower bound. Koval and Kwan prove that there exists \(c>0\) such that, for all sufficiently large \(n\), \[ \#\{\text{unlabelled }n\text{-vertex DS graphs}\}\ge e^{cn}, \] and obtain \(c=0.01\) for large \(n\). They also prove an analogous exponential lower bound for graphs determined by their signless Laplacian spectrum. This improves the earlier \(e^{c\sqrt n}\)-type lower bounds coming from partition-enumerated DS families [2309.09788]. Their construction uses three spectral lenses: Laplacian, signless Laplacian, and adjacency spectra. First, certain “nice” sun-like graphs are shown to be determined by their Laplacian spectrum by combining Kirchhoff’s matrix-tree theorem with weighted spectral moments. Second, for \((\ell,k)\)-nice graphs with \(\ell\equiv 2\pmod 4\), the signless Laplacian spectrum forces bipartiteness via the divisibility properties of the product of nonzero \(|L|\)-eigenvalues. Third, the signless Laplacian spectrum is transferred to the adjacency spectrum through line graphs, after excluding generalized line graph impostors by determinant calculations. Dirichlet’s theorem on primes in arithmetic progressions then yields exponentially many pairwise non-isomorphic DS graphs [2309.09788]. An algorithmic approach studies the weaker property DGS, meaning determined by the generalized spectrum \[ S_G=(\operatorname{Spec}(A(G)),\operatorname{Spec}(A(\overline G))). \] Here DS implies DGS, but not conversely. For controllable or almost controllable graphs, generalized cospectral mates are encoded by rational regular orthogonal matrices \(Q\) satisfying \[ Q^\top A(G)Q=A(H),\qquad Qe=e. \] The search is reduced to congruence constraints derived from walk matrices, Smith normal forms, discriminants, and the level \(\ell(Q)\), and then to maximum cliques in an auxiliary graph \(\Omega(G)\). In one experiment on 10,000 randomly generated graphs on 50 vertices, at least 9945 were DGS. This is evidence for spectral rigidity, but it is evidence for DGS rather than a proof of the DS conjecture [2207.10540]. The present status is therefore asymmetric: the asymptotic DS conjecture remains open, yet both constructive and algorithmic evidence indicate that spectrally rigid graphs are much more abundant than earlier proofs could show [2309.09788] [2207.10540]. ## 5. Haemers bounds, quantum capacity, and related conjectural landscapes In information-theoretic graph theory, Haemers’ name is attached less to a conjecture than to a family of upper bounds on Shannon capacity. For a graph \(G\), the Haemers bound over a field \(F\) is \[ H^F(G):=\min\{\operatorname{rank}_F(M): M_{gg}\ne 0,\; M_{gg'}=0 \text{ for nonedges}\}, \] and the fractional Haemers bound is \[ H_f^F(G):=\inf\{\operatorname{rank}(M)/d:\;M\in M_F^d(G)\}. \] These satisfy \[ \Theta(G)\le H_f^F(G)\le H^F(G), \] where \(\Theta(G)\) is the Shannon capacity [1810.00744]. Quantum asymptotic spectrum methods show that the fractional real and complex Haemers bounds are spectral points for the quantum cohomomorphism preorder: \[ H_f^\mathbb R,\;H_f^\mathbb C\in X(G,\le_q), \qquad H_f^\mathbb R(G)=H_f^\mathbb C(G). \] Consequently, \[ \Theta_q(G)\le H_f^\mathbb R(G)=H_f^\mathbb C(G), \] so real and complex fractional Haemers bounds upper bound the quantum Shannon capacity. By contrast, for certain finite fields \(F_p\), the corresponding finite-field Haemers bounds need not be monotone for \(\le_q\) and can be strictly smaller than \(\Theta_q(G)\) [1810.00744]. These results separate quantum capacity from Lovász’s theta function. Since Haemers had already exhibited graphs with \(H^\mathbb R(G)<\theta(G)\), the quantum paper deduces that \[ \Theta_q(G)\le H^\mathbb R(G)<\theta(G) \] for suitable \(G\). Hence the quantum Shannon capacity does not coincide with \(\theta(G)\). As a consequence, the two conjectures \[ \Theta_*(G)=\theta(G)\quad\text{for all }G \] and \[ \Theta_q(G)=\Theta_*(G)\quad\text{for all }G \] cannot both be true [1810.00744]. A further extension is the tracial Haemers bound \(H_{tr}(G)\), defined through projections in a von Neumann algebra with a normal tracial state. It upper bounds the commuting quantum independence number and is multiplicative under the strong product: \[ H_{tr}(G\boxtimes H)=H_{tr}(G)\,H_{tr}(H). \] Thus it is an upper bound on \(\Theta_{qc}(G)\). The tracial and fractional Haemers bounds are linked to operator-algebraic foundations: if there exists a graph \(G\) with \[ H_{tr}(G)