The main result is an explicit optimal embedding of cluster graphs on $n$ vertices, which is strongly NP-complete and admits no FPTAS, while the farthest alignment is polynomial-time solvable.
Abstract
The cluster graphs on $n$ vertices, the disjoint unions of complete graphs, have the integer partitions of $n$ as their isomorphism classes, and the quotient edit distance $q^*(\lambda,\mu)=\min_{\sigma\in S_n}|E(G_\lambda)\triangle\sigma E(G_\mu)|$ makes that set a metric space. Its geometry and its complexity both issue from one identity: $q^*$ is an affine function of the maximum of $\lVert X\rVert_F^2$ over the contingency tables with margins $\lambda$ and $\mu$. Our main result is an explicit optimal embedding. The weighted dyadic sums of the Ferrers staircase, taken at the critical exponent $\frac14$, give a map $F_n$ into $\ell_2^{\,<4n}$ that acts on a single partition and is computable in $O(n)$ time, and its distortion is $\Theta(n^{1/4})$. That order is optimal, since $c_2(\mathcal K_n)=\Theta(n^{1/4})$: the lower half follows from a $\Theta(\sqrt n)$-dimensional Hamming cube of partitions and Enflo's theorem, so the determination needs no other external input. The analytic core is a scale-free inverse inequality for every integer sequence with $v(1)=v(N+1)=0$ and $v(s)-v(s+1)\in s\mathbb Z$: its critical dyadic energy is at least $\lVert v\rVert_1^2/(63504\sqrt{\mathrm{TV}(v)})$. Combinatorially the same identity yields two explicit $\ell_1$ models, the vertex-mass metric on sorted degree sequences with $\frac12\delta_1\le q^*<\frac32\delta_1$ and the block-energy metric with $q^*\le B\le2q^*-1$, both constants optimal; hence $c_1(\mathcal K_n)\le2$, and an $O(n\log n)$-time algorithm returns an alignment of cost below $2q^*$ carrying the certificate $q^*\in[\lceil(B+1)/2\rceil,B]$. Computationally, deciding $q^*(\lambda,\mu)\le Q$ is strongly NP-complete and admits no FPTAS, while the farthest alignment is polynomial-time solvable. The best constant in the inverse inequality remains open; an exactly solvable chirp family caps it at $\frac23$.
Let $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$ be the eigenvalues of a simple graph $G$ of order $n$. The HL-index of $G$ is defined by $R(G)=\max\|\lambda_h|,|\lambda_\ell|\}$ with $h=\lfloor(n+1)/2\rfloor$ and $\ell=\lceil(n+1)/2\rceil$.In this paper, we prove that if $G$ is $ K_4$-minor-free or $ K _ {2,3} $-minor-free, then $R(G)\leq\sqrt{5}-1$ with equality attained by an infinite family of outerplanar graphs.Moreover, we show that $R(G)\leq\sqrt{d-2}$ for triangle-free graphs with maximum degree at most $d$ and average degree at most $(d-2)(d^2-2d+2)/(d^2-3d+5)$.
For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche--Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.
For an integer $k\ge2$, let $\lambda_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $\gamma(r)$ due to Der\k{e}gowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of K\"onig and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.
A de Bruijn sequence is the cyclic prototype of a Cayley-graph observation problem: when does the ordered label word on a translated window $gY$ determine the vertex $g$? We distinguish three parameters. The unrestricted number $\operatorname{sep}_q(G)$ minimizes an arbitrary separating pattern; the connected number $\operatorname{csep}_q(G,S)$ requires a connected Cayley window containing $Y_S=\{1\}\cup S$; and the one-step number $\chi_1(G,S)$ fixes $Y_S$ and minimizes the alphabet. Thus $\operatorname{sep}_q$ is a group-level baseline, $\operatorname{csep}_q$ measures the cost of locality, and $\chi_1$ tests the smallest prescribed local window. The organizing theme is the tension between information and locality. Carbon tori test the gap between $\operatorname{sep}_q$ and $\operatorname{csep}_q$: for generalized dihedral groups $\mathbb{F}_{\ell^d}^{\times}\rtimes C_2$ we prove, for odd prime powers $\ell$, the sharp baseline $\operatorname{sep}_\ell=d+1$ and construct connected zig-zag windows, while the order-$14$ Heawood torus satisfies $\operatorname{sep}_4=2$ and $\operatorname{csep}_4=4$. The spherical $A_5$ example and a finite simple-group comparison test the fixed one-step window: explicit symmetric cubic generating tuples give $\chi_1(A_5,S)=3$ and $\chi_1(\operatorname{PSL}_2(\mathbb{F}_7),S)=4$, both at the counting bound, with structured matrix-coefficient certificates. Cyclic-coset packings, finite-field coordinates, and restricted matrix coefficients are used only as the construction tools these two examples require.
This work develops an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$, and shows that matching results hold for other random models with negatively associated entries.
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
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