Papers

The Hamming distance between two equal length words $\alpha, \beta$ is the number of positions where $\alpha$ and $\beta$ differ. For $x, y \in \Sigma^*$ and antimorphic involution $\theta$, $x$ $\theta$-commutes with $y$, if the Hamming distance between $xy$ and $\theta(y)x$ is zero. When the Hamming distance between $xy$ and $\theta(y)x$ attains its minimum non-zero value of one, then we say $x$ nearly $\theta$-commutes with $y$. This manuscript investigates properties of $x$ and $y$ such that the Hamming distance between $xy$ and $\theta(y)x$ is one. We provide a complete characterization of such $x$ and $y$. We introduce a binary relation $R_\theta$ on $\Sigma^*$, where $x R_\theta y$ holds if and only if $x$ nearly $\theta$-commutes with $y$. Finally, for a given $y$, we collect all $x$ such that $x$ nearly $\theta$-commutes with $y$, and discuss various properties of this set.

Product structure theory aims to understand complex graphs by embedding them into products of simpler graphs. In this direction, Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2022) put forth the conjecture that all graphs of degree-$d$ polynomial growth (i.e., where balls of radius $r$ have $\mathcal{O}(r^d)$ vertices) can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size clique. In this paper, we disprove this conjecture for $d = 4$. The counterexamples are finite subgraphs of a Cayley graph of the discrete $3$-dimensional Heisenberg group $\mathbb{H}(\mathbb{Z})$. These graphs were first proposed by Huang and McCarty as potential counterexamples to the conjecture. A key technical tool of our proof is the ”quantitative central collapse” theorem due to Cheeger, Kleiner and Naor (2011), guaranteeing that every Lipschitz map from the continuous Heisenberg group $\mathbb{H}$ to the function space $L_1$ collapses along a central line.

The chromatic threshold of Erdős and Simonovits asks when a minimum-degree condition forces every \(H\)-free graph to have bounded chromatic number. Thomassen's homomorphism threshold strengthens this by requiring a bounded \(H\)-free homomorphic image. The recently introduced blowup threshold \(\delta_{\mathrm B}(H)\) asks for a still more rigid conclusion: when must every sufficiently dense maximal \(H\)-free graph be an actual blowup of a bounded graph? Thus the blowup threshold measures when quotient-level structure can be upgraded to exact bounded-template structure. We show that, although chromatic and homomorphism thresholds are often hard to separate, the stronger blowup threshold diverges from the chromatic threshold in several fundamental ways. First, we prove that \(\delta_{\mathrm B}(H)>0\) for every non-bipartite graph \(H\). Hence, unlike the chromatic threshold, the blowup threshold never vanishes outside the bipartite world. Second, we prove that $\delta_{\mathrm B}$ is not monotone under taking induced subgraphs. This shows that the blowup threshold is sensitive to global features of the forbidden graph and cannot be classified by a direct analogue of the monotonicity-based strategy used for chromatic thresholds. Third, we prove that $\delta_{\mathrm B}(H)=\frac{1}{4}$ for a natural family of \(3\)-chromatic constrained blowups of odd cycles. This gives a new exact blowup-threshold value beyond the chromatic-threshold spectrum.

We give two general constructions for $2$-designs, that can be used recursively, and interchangeably, to produce new infinite families of $2$-designs admitting block-transitive groups of automorphisms which preserve arbitrarily large posets of partitions of the point-set. The only arbitrarily large posets for which constructions were previously known are chains of arbitrary length. Using the constructions we exhibit new infinite families of poset-imprimitive block-transitive $2$-designs corresponding to several different arbitrarily large posets, as well as constructions for most posets with four nodes.

A set system $\mathcal{F}$ is called $t$-intersecting if $|A\cap B|\ge t$ for every pair of sets $A,B\in \mathcal{F}.$ A set system $\mathcal{F}$ is $k$-Sperner if it does not contain a chain of length $k+1$. Balogh, Linz and Patkós (Combinatorial Theory, 2023) conjectured an extremal result for $t$-intersecting $k$-Sperner families when $n+t$ is odd. In this note we give an explicit construction that is $t$-intersecting and $k$-Sperner, and whose size exceeds that of the conjectured fixed-star construction for infinitely many values of $n$. Consequently, we disprove the Balogh-Linz-Patkós conjecture for all $t\ge 2$ and $k\ge 2$ satisfying $k(t-1)\ge t+1$.

Let $G$ be a group and $K$ be a normal subgroup of $G$. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed, and the order is disregarded. A product-$K$ sequence is a sequence whose terms can be ordered such that their product in $G$ belongs to $K$. The set $\mathcal B_K (G)$ of all product-$K$ sequences over $G$ forms a monoid, called the monoid of product-$K$ sequences, under the operation of sequence concatenation. In this paper, we investigate the algebraic and arithmetic properties of the monoid $\mathcal B_K (G)$. Among our main results, we provide precise characterizations of when the monoid $\mathcal B_K (G)$ satisfies key properties, namely being a (transfer) Krull, seminormal, or (half-)factorial. Our results generalize existing frameworks, making them applicable to both the classical abelian and the more recently developed non-abelian settings.

The Ramsey community number $r_\kappa$ is the smallest network size at which a graph is better described by a partition into communities than by no partition, under a prescribed detection rule. On a scale-free graph this question is confounded: a block model can split the network merely to absorb its degree distribution. I compute $r_\kappa$ analytically for the deterministic pseudofractal scale-free web of Dorogovtsev, Goltsev, and Mendes, separating genuine community structure from degree heterogeneity with two closed-form detection rules. Under a plain Bernoulli stochastic block model, the web's natural recursive bipartition is unpreferred while small and breaks at $r_\kappa=1095$ nodes, with a log-evidence growing as $(\ln 3-\tfrac{2}{3}\ln 2)n$. Under a degree-corrected model tested against the configuration-model null, the same partition survives, breaking far earlier at $r_\kappa=42$, with a log-evidence growing as $(2\ln 3-\tfrac{4}{3}\ln 2)n$ – exactly twice the plain slope, and independent of the prior. Degree correction reverses the ordering of the candidate cuts, demoting the hub-leaf split and elevating the recursive one. Because the web is self-similar, the best description is not two communities but a nested hierarchy: the degree-corrected evidence keeps rising as the partition is refined, and is maximised at of order $\sqrt{n}$ communities of $\sim\sqrt{n}$ nodes. A purely local recursive rule thus builds true hierarchical community structure, over and above the scale-free degree sequence it also produces, in an exactly solvable setting.

For a graph $\Gamma$, the splitting field of $\Gamma$ is defined as the splitting field of the characteristic polynomial of $\Gamma$ over rationals. The algebraic degree of $\Gamma$ is defined by the extension degree of its splitting field over rationals. Let $k$ be a positive integer. We call a finite group $G$ \textit{Cayley $k$-integral} if, for every inverse-closed subset $S$ of $G$, the algebraic degree of the Cayley graph $\Cay(G,S)$ does not exceed $k$. We give a complete classification of all finite Cayley $2$-integral groups. It is shown that a finite abelian group is Cayley $2$-integral if and only if it is isomorphic to one of the following forms: $G \cong \mathbb{Z}_2^r \times \mathbb{Z}_5^s$, $\mathbb{Z}_2^r \times \mathbb{Z}_4^s \times \mathbb{Z}_8^t$, or $\mathbb{Z}_2^r \times \mathbb{Z}_3^s \times \mathbb{Z}_{12}^t$, where $r, s, t \geq 0$. Furthermore, we prove that the set of finite non-abelian Cayley $2$-integral groups consists of the infinite family $Q_8 \times \mathbb{Z}_2^n$, with $n \geq 0$, and $22$ specific groups.

Problems complete for the existential theory of the reals ($\exists \mathbb{R}$) arise throughout discrete geometry. We introduce satisfiability modulo realizability, a SAT-based approach for solving satisfiable instances of $\exists \mathbb{R}$ whose solutions correspond to realizable geometric configurations. Our method encodes an underapproximation of a geometric problem as a SAT instance over abstract order types. Since almost all abstract order types are unrealizable, naive search is infeasible. We guide the search toward realizable order types using diversity-driven sampling, partial realizability feedback, and a novel flippability heuristic that passes only limited information between components. We apply our method to discrete geometry problems and resolve an open problem by showing that the largest set of points avoiding empty convex hexagons and convex heptagons is of size 23.

In this paper, we consider connected signed graphs with smallest eigenvalue at least $-3-\varepsilon$ for a small positive constant $\varepsilon$. We prove that if such a signed graph has sufficiently large minimum valency, then its smallest eigenvalue is at least $-3$, and the lattice associated with it, which is generated by squared norm $3$ vectors, is a sublattice of a direct sum of the standard lattice $\mathbb{Z}^n$ and copies of the root lattice $E_8$. Moreover, there exist infinitely many connected signed graphs with smallest eigenvalue at least $-3$ containing it as a proper induced subgraph. Furthermore, we discuss signed graphs with smallest eigenvalue $-3$ arising from rootless irreducible unimodular lattices.

We study two inverse problems for shortest-path metrics of Okamura-Seymour instances: recognizing metrics realizable by outerplanar graphs, and computing minimum-edge Okamura-Seymour realizations. We introduce the notion of repelling pairs, a metric certificate that the shortest paths corresponding to two terminal pairs must be vertex-disjoint in every realization. Our central structural result is that, for an Okamura-Seymour metric with a prescribed cyclic order, a terminal path structure in an Okamura-Seymour instance can be realized by nonnegative edge lengths if and only if the paths assigned to every repelling pair are vertex-disjoint. Building on the notion of repelling pairs, we give algorithmic answers to the inverse problems. First, we design an algorithm that, given a metric, decides in polynomial time whether or it admits an outerplanar realization and constructs one when one exists. Second, given an Okamura-Seymour metric, we efficiently compute a canonical medial template whose crossing number equals the minimum number of edges in any Okamura-Seymour realization. The minimum-edge graph structures are exactly the primal graphs of arrangements of this template, and each can be assigned realizing edge lengths in polynomial time.

We give explicit counterexamples to two rearrangeability conjectures for shuffle-type networks. First, for every $N\ge2$ we construct a simple $N$-regular ordered two-stage graph $L_N$ with $F(L_N)=2$ and $R(L_N)\ge N$, refuting the graph-theoretic Beneš inequality $R(L)\le2F(L)$ and its partition-stabilizer form as stated on Open Problem Garden. We retain the sharp cut obstruction, exact mask-composition identity, exact middle criterion, first nontrivial-level result, and balanced-middle sufficient condition that explain which extra hypotheses can replace mere external connectivity. Second, for the standard directed shuffle-exchange network, we prove $d(k,3)=6$ for every $k\ge3$, while the known binary value is $d(2,3)=5$. Hence the shuffle-exchange conjecture $d(k,n)=2n-1$ fails already at $(k,n)=(3,3)$, and the remaining upper bound $d(k,n)\le3n-3$ for $k\ge3$ suggests $d(k,n)=3n-3$ as a natural replacement problem.

Let $A(k)$ be the largest possible number of moves in a north-east lattice path whose visited vertices contain no $k$ collinear points. Gerver (1979) and Gerver and Ramsey (1979) gave lower and upper bounds on $A(k)$ of the form \[ \exp\left(\Omega(\log(k)^2)\right)\le A(k)\le \exp\left(O(k^4)\right). \] Improving upon these results, we show that \[ \exp\left(\Omega(k^{1/3})\right)\le A(k)\le \exp\left(\left(\frac{2}{e}+o(1)\right)(k-1)^2\right). \]

Interlacing of the real roots of a weighted matching polynomial for a graph $G$ and that of a vertex-deleted subgraph is classical and well-known. In the context of strict interlacing of distinct roots, a demonstrated graph construction gives rise to a new classification of graphs, called ${ \rm SRSI}$ graphs, which include graphs that contain a Hamilton path. Graphs with a perfect (or nearly perfect) matching are shown to exhibit the SRSI$_w(v)$ property with respect to a particular weighting and for specific vertices, and all such graphs are characterized via this graph construction. As a consequence, we also characterize the trees that possess the SRSI property for all edge weightings and all vertices.

For a delta-matroid, the maximum twist width theorem states that the maximum width over all twists can be reached along a non-decreasing sequence of intermediate twist widths. In this paper we study analogous monotone maximum width sequences for partial twualities generated by twist and loop complementation. We prove that, for each non-twist partial-twuality operation on a vf-safe delta-matroid, there exists a subset attaining the corresponding maximum partial-twuality width whose elements can be ordered so that the successive intermediate widths are non-decreasing. Together with the known twist case, this gives a monotone maximum width theorem for all five nontrivial partial-twuality operations on vf-safe delta-matroids. We also prove feasible-set attainment results for the operations $\ast\times$ and $\ast\times\ast$. Finally, we translate these results to ribbon graphs, obtaining monotone sequences for maximum partial-twuality Euler genera and spanning quasi-tree attainment results for the corresponding ribbon graph operations.

Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of matroid states, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.

We prove that, for every fixed integer $g\geq 1$, the largest cardinality of a $B_2[g]$ subset of the first $n$ squares is at least a positive constant, depending only on $g$, times $$ n^{\frac{2g}{2g+1}}(\log n)^{\frac{2-2^g}{2g+1}}, $$ for all sufficiently large $n$. For $g=1$, this recovers the theorem of Lefmann and Thiele on Sidon subsets of the first squares. The proof follows their hypergraph method, but replaces the $4$-uniform hypergraph encoding two representations as a sum of two squares by a $2(g+1)$-uniform hypergraph encoding $g+1$ such representations. The main point is to verify that this higher-uniformity hypergraph has few edges and few $2$-cycles; the lower bound then follows from an independence theorem for uncrowded hypergraphs due to Duke, Lefmann and Rödl.

We present an embedded (in $G/B$) degeneration of Lusztig varieties (which generalize type $A$ Hessenberg varieties) to certain unions of Richardson varieties, giving a simultaneous reproof (and extension) of results of Anderson–Tymoczko, Harada–Horiguchi–Masuda–Park, and Kim. Although torus-equivariant, the degeneration is not Gröbner. In the case that the Lusztig variety is the permutahedral toric variety, this degeneration provides a subdivision of the permutahedron into Bruhat interval polytopes, and we prove a more general result showing equivariant degenerations of projective toric varieties produce subdivisions of the moment polytope (as was shown in the Gröbner case by Sturmfels). A Gröbner degeneration would result in a regular subdivision, and despite our degeneration not being Gröbner we show in types $A,B,C$ that our subdivisions of the permutahedron are indeed regular.

For a graph $G$ with line graph $L(G)$, $\chi(L(G)^2)$ and $\omega(L(G)^2)$ are called the strong chromatic index and strong clique index of $G$, respectively. A well-known conjecture of Erdős and Nešetřil (1985) posits that $\chi(L(G)^2)\le \frac{5}{4}\Delta(G)^2$. Related to that, Faudree, Gyárfás, Schelp and Tuza (1990) conjectured that $\omega(L(G)^2) \le \frac{5}{4}\Delta(G)^2$. We show that $\omega(L(G)^2) \le \frac{2607}{1987}\Delta(G)^2 < \frac{21}{16}\Delta(G)^2$ improving the upper bound $\frac{4}{3}\Delta(G)^2$ of Faron and Postle. Indeed, we make progress towards a stronger conjecture of Faron and Postle in terms of Ore-degree. For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ denote the smallest integer such that any graph $G$ with size at least $h_t(\Delta)$ and maximum degree $\Delta(G)\le \Delta$, contains two edges with distance at least $t$. An old problem of Erdős and Nešetřil (1986) concerns estimating the quantity $h_t(\Delta)$ and can be thought of as the edge-version of the degree-diameter problem. Chung, Gyárfás, Tuza and Trotter established the sharp inequality $h_2(\Delta)\le \frac{5}{4}\Delta^2+1$. We disprove two conjectures of Cambie, Cames van Batenburg, Joannis de Verclos and Kang concerning the next open case $h_3(\Delta)$.

A 2-partition of a digraph is a partition of its vertex set into two nonempty parts. Degree-constrained 2-partition problems are generally computationally difficult, even when the prescribed properties are expressed only in terms of minimum indegree, minimum outdegree, or minimum semidegree. Bang-Jensen and Christiansen~\cite{B-C} conjectured that the minimum-degree partition problems would be polynomial-time solvable on semicomplete digraphs when the degree thresholds are fixed, and Bang-Jensen and Gutin~\cite{B-G-Classes} posed the related Problems~2.8.15 and~2.8.16. We resolve this conjecture. More precisely, for every fixed pair of integers $k_1,k_2\ge 2$, we give deterministic polynomial-time algorithms that decide whether a given semicomplete digraph admits a $(\delta^+\geq k_1,\delta^-\geq k_2)$-partition, a $(\delta^+\geq k_1,\delta^0\geq k_2)$-partition, or a $(\delta^0\geq k_1,\delta^0\geq k_2)$-partition, and construct such a partition whenever one exists. Here, $\delta ^+,\delta ^-,\delta ^0$ represent the minimum out-, in-, semi-degree, respectively. The algorithms use small degree certificates, minimal cores, closure and protective-set arguments, and deterministic universal colorings with monotone recoloring, which develop a new method in partition algorithm construction.

A poset $P$ is said to satisfy the finite antichain condition, or FAC for short, if it has no infinite antichain. Such posets exhibit rich and complex structure, and it was conjectured by Aharoni and Korman in 1992 that any FAC poset $P$ possesses a chain $C$ and a partition into antichains such that $C$ meets every antichain of the partition. While this conjecture is now known to be false, in this paper we prove that the conjecture does hold true for a broad class of posets. In particular, we prove that the Aharoni–Korman conjecture holds for countable posets containing no saturated chain $D$ such that either $D$ or its reverse $D^*$ is of the form $\bigoplus_{x\in\omega} D_x$, where each $D_x$ is infinite and co-wellfounded. In pursuit of this goal, we prove several structural results, the foremost of which demonstrates how a countable FAC poset may be broken up into a collection of scattered posets which reflect the structure of the poset as a whole.

The Erdős–Gallai theorem states that every graph of average degree $d$ contains a cycle of length at least $d$. We prove the following robust extension of the Erdős–Gallai theorem: For every $c>0$ there exists $K$ such that for all $d\geq K$, $p\geq K/d$ and every graph $G$ with average degree $d$, the random graph $G_p$ obtained by independently sampling each edge of $G$ with probability $p$ contains a cycle of length at least $(1-c)d$ asymptotically almost surely as $|V(G)|\to \infty$. With related methods, we prove the following hyperstability version of the Erdős–Gallai theorem: any graph $G$ without a cycle of length at least $d$ is at most $c d|V(G)|$ edge deletions away from a graph all of whose connected components have a vertex-cover of size at most $d$. At the core of our argument lies a very general structure theorem about graphs that originates from results of Pokrovskiy concerning the hyperstability of bounded-degree trees.

We study the Sylow permutation character of the symmetric group at the prime 2 and prove some new bounds on the number of parts of partitions corresponding to its constituents.

A conjecture posed by Ohsugi and Tsuchiya (2019) postulates that the Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive. We disprove this conjecture by exhibiting an infinite family of counterexamples. The smallest example provided by our construction is a $36$-dimensional symmetric edge polytope.

We study characterisations of strong $\Delta$-matroids, compiling a list of five equivalent descriptions. We show a variant of Wenzel's exchange property and the hyperplane exchange property of Borovik-Gelfand-White are equivalent. We also introduce two novel characterisations in terms of 'peerless' and 'isolated' antipodes within the system of feasible sets, banning certain configurations of antipodes either globally or locally. As a corollary, we obtain new 'local' exchange axioms for matroids and $\Delta$-matroids. We give algebraic motivation for these new characterisations by introducing the peerless antipode equations, tropical equations that govern whether a $\Delta$-matroid has no peerless antipodes. We show that these arise as the tropicalisation of a specific basis of quadratics cutting out the orthogonal Grassmannian.

We introduce a natural Markov chain on chord diagrams, which, at every step, selects two random chords and swaps them if doing so preserves the diagram's genus. This generalizes the chord swap chain on the Catalan structure of non-intersecting chord diagrams. We show that for fixed genus, the chain mixes in polynomial time.

We design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. Our key observation is that the cap-volume function of such a measure, i.e., the volume cut off by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This lets us recast prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension, this yields polynomial-time algorithms to decide the existence of cuts, describe the full solution set, and sample or enumerate solutions. We extend this framework to the center transversal theorem, showing that spaces of deep affine flats are semialgebraic, which holds for centerpoints. We further show that the set of centerpoints of a convex polytope coincides with its floating body at level $1/(d+1)$, a useful semialgebraic description.

We prove an extremal theorem for positive Ollivier/Lin–Lu–Yau curvature: every graph of order \(n\geq 8\) with more than \[ T(n)=\frac{n^2-3n}{2}-\left\lceil\frac{n}{2}\right\rceil+2 \] edges has positive Ollivier/Lin–Lu–Yau curvature, and this threshold is optimal. Moreover, for even $n\geq 12$, there exists a unique graph with $T(n)$ edges that has an edge with non-positive curvature. For $n=8,10$ and odd $n\geq 9$, the extremal graphs are not unique. This suggests a new class of extremal graph-theoretic problems arising from discrete curvature notions.

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H\subseteq G$ is $\Delta(G)$-overfull if $|E(H)|>\Delta(G)\left\lfloor |V(H)|/2\right\rfloor$. In any edge coloring of $G$, each color class restricted to $H$ is a matching of size at most $\left\lfloor |V(H)|/2\right\rfloor$. Thus, if $G$ contains a $\Delta(G)$-overfull subgraph, then $G$ cannot be edge-colored with only $\Delta(G)$ colors. By Vizing's Theorem, $\chi'(G)\le \Delta(G)+1$, and hence $G$ is class $2$. In 1986, Chetwynd and Hilton conjectured that whenever $\Delta(G)>|V(G)|/3$, the converse also holds: every class $2$ graph $G$ contains a $\Delta(G)$-overfull subgraph. This statement, commonly known as the Overfull Conjecture, is one of the most influential conjectures in graph edge coloring. It would imply a polynomial-time algorithm for determining the chromatic index of graphs $G$ with $\Delta(G)>|V(G)|/3$, and would also imply several other longstanding conjectures in the area, including the Just-overfull Conjecture and the Vertex-splitting Conjecture. In previous work, the third author verified the conjecture for large graphs $G$ with maximum degree at least $13|V(G)|/14$. In this paper, we confirm the conjecture for robust expanders satisfying certain density constraints. As a consequence, for every $0<\varepsilon<1$, the conjecture holds for all sufficiently large graphs $G$ with maximum degree at least $(1+\varepsilon)|V(G)|/2$.

We develop a tensor-amplification framework for Sidorenko-type inequalities in graphon classes. The framework applies to any admissible class, meaning a class closed under tensor powers and normalized principal restrictions. These two closure properties isolate the structural input needed for the amplification arguments, while preserving natural positivity constraints such as the doubly nonnegative constraint. For every admissible class $\mathcal{C}$, we prove two transfer principles. First, equality cases regularize optimally: if a non-matching graph $H$ is $\mathcal{C}$-Sidorenko, then every equality case $t(H,W)=p(W)^{e(H)}$ with $W\in\mathcal{C}$ is regular. Consequently, relative forcing is equivalent to relative regular-forcing for every non-matching $\mathcal{C}$-Sidorenko graph. Second, in the range $v(H)\le e(H)$, ordinary $\mathcal{C}$-Sidorenko is equivalent, as a universal property over $\mathcal{C}$, to the spectral inequality $t(H,W)\ge \rho(W)^{2e(H)-v(H)}p(W)^{v(H)-e(H)}$ for every non-zero $W\in\mathcal{C}$. The spectral transfer is obtained from a Perron-biased tensor regularization theorem detecting the Perron spectral radius on the exponential scale. We also prove quantitative near-equality variants and apply the framework to doubly nonnegative graphons and bounded doubly nonnegative kernels. This yields spectral equivalences for Sidorenko-good graphs in the range $v(F)\le e(F)$, and identifies Sidorenko-good forcing with regular-KNRS forcing for non-matching Sidorenko-good graphs.