We describe the computer-aided classification of equitable partitions of the $12$-cube with quotient matrix $[[2,10],[6,6]]$, or, equivalently, simple orthogonal arrays OA$(1536,12,2,7)$, or order-$7$ correlation-immune Boolean functions in $12$ variables with $1536$ ones (which completes the classification of unbalanced order-$7$ correlation-immune Boolean functions in $12$ variables). We find that there are $103$ equivalence classes of the considered objects, and there are only two almost-OA$(1536,12,2,8)$ among them. Additionally, we find that there are $40$ equivalence classes of pairs of disjoint simple OA$(1536,12,2,7)$ (equivalently, equitable partitions of the $12$-cube with quotient matrix $[[2,6,4], [6,2,4], [6,6,0]]$) and discuss the existence of a non-simple OA$(1536,12,2,7)$.
Keywords: orthogonal arrays, correlation-immune Boolean functions, equitable partitions, perfect colorings, intriguing sets.
It is proved that the vertex set of any simple graph $G$ can be equitably partitioned into $k$ subsets for any integer $k\geq\max\{\big\lceil\frac{Δ(G)+1}{2}\big\rceil,\big\lceil\frac{|G|}{4}\big\rceil\}$ so that each of them induces a linear forest.
The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with independent crossings has an equitable partition into $m$ induced forests for any $m\geq 8$. Moreover, we decrease this lower bound 8 for $m$ to 6, 5, 4 and 3 if we additionally assume that the girth of the considering graph is at least 4, 5, 6 and 26, respectively.
Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several well-known eigenvalue bounds. In this work we aim to further our algebraic and computational understanding of weight-equitable partitions. We do so by showing several spectral properties and algebraic characterizations, and by providing a method to find coarse weight-equitable partitions.
An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\ldots,G[V_k])$ of $G$ such that $(V_1,V_2,\ldots,V_k)$ is a partition of $V(G)$ and $-1\le |V_i|-|V_j|\le 1$ for all $1\le i<j\le k$. We prove that every planar graph admits an equitable $2$-partition into $3$-degenerate graphs, an equitable $3$-partition into $2$-degenerate graphs, and an equitable $3$-partition into two forests and one graph.
We study equitable partitions of Latin-square graphs, and give a complete classification of those whose quotient matrix does not have an eigenvalue $-3$.
The eigenvalues of the Hamming graph $H(n,q)$ are known to be $λ_i(n,q)=(q-1)n-qi$, $0\leq i \leq n$. The characterization of equitable 2-partitions of the Hamming graphs $H(n,q)$ with eigenvalue $λ_{1}(n,q)$ was obtained by Meyerowitz in [15]. We study the equitable 2-partitions of $H(n,q)$ with eigenvalue $λ_{2}(n,q)$. We show that these partitions are reduced to equitable 2-partitions of $H(3,q)$ with eigenvalue $λ_{2}(3,q)$ with exception of two constructions.
In this paper, we present an equitable partition theorem of tensors, which gives the relations between $H$-eigenvalues of a tensor and its quotient equitable tensor and extends the equitable partitions of graphs to hypergraphs. Furthermore, with the aid of it, some properties and $H$-eigenvalues of the generalized power hypergraphs are obtained, which extends some known results, including some results of Yuan, Qi and Shao.
We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.
We study equitable 2-partitions of the Johnson graphs J(n,w) with a quotient matrix containing the eigenvalue lambda_2(w,n) = (w-2)(n-w-2)-2 in its spectrum. For any w>=4 and n>=2w, we find all admissible quotient matrices of such partitions, and characterize all these partitions for w>=4, n>2w, and for w>=7, n = 2w, up to equivalence.
We show that every ridge unfolding of an $n$-cube is without self-overlap, yielding a valid net. The results are obtained by developing machinery that translates cube unfolding into combinatorial frameworks. Moreover, the geometry of the bounding boxes of these cube nets are classified using integer partitions, as well as the combinatorics of path unfoldings seen through the lens of chord diagrams.
For d-regular graph G, an edge-signing sigma:E(G) \rightarrow {-1,1} is called a good signing if the absolute eigenvalues of adjacency matrix are at most 2 \sqrt{d-1}. Bilu-Linial conjectured that for each regular graph there exists a good signing. In this paper, by using new concept "Equitable Partition", we solve the Bilu-Linial Conjecture for some cases. We show that how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph G, then we can find a good signing for a 2-lift of G.
Equitable partitions of regular graphs, and perfect sets in normal Cayley graphs
School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings. In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching--covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the p…
With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings of a Hypercube, Sib. Math. J. 48 (2007), 740-745] as an equitable partition of the $13$-cube with quotient matrix $[[0,13],[3,10]]$. By shortening the OA$(1536,13,2,7)$, we obtain $3$ inequivalent orthogonal arrays OA$(768,12,2,6)$, which is a complete classification for these parameters too. After our computing, the first parameters of unclassified binary orthogonal arrays OA$(N,n,2,t)…
We suggest a combinatorial method of encoding continuous symbolic dynamical systems. A~continuous phase space, the infinite-dimensional cube, turns into the path space of a tree, and the shift is mapped to a transformation which was called a "transfer." The central problem is that of distinguishability: does the encoding separate almost all points of the space? The main result says that the partition of the cube into Weyl simplices satisfies this property.\footnote{{\it Keywords:} combinatorial encoding, transfer, Bernoulli scheme, graded graph.
Let $π=\{V_1,\dots,V_r\}$ be an equitable partition of the vertex set of a directed graph (digraph) $X$. It is well known that the characteristic polynomial $ϕ(X/π,x)$ of a quotient graph $X/π$ divides that of $X$, but the remainder part is not well investigated. In this paper, we define a deletion graph $X\backslashπ$ over an equitable partition $π$, which is a signed directed graph defined for a fixed set of deleting vertices $\{\bar{v}_i\in V_i, i=1,\cdots,r\}$, and give a similarity transformation exchanging the adjacency matrix $A(X)$ which is compatible with the equitable partition for a…
We study how the number $c(X)$ of components of a graph $X$ can be expressed through the number and properties of the components of a quotient graph $X/\sim.$ We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of component equitable partition, exhibiting interesting inclusions among the various classes of considered homomorphisms. As a consequence, we find a procedure for computing $c(X)$ when the projectio…
Quantum fractional revival on zero-divisor graphs over $\mathbb{Z}_n$
Mahidol University（玛希隆大学）；Centre of Excellence in Mathematics, MHESI（数学卓越中心，高等教育委员会）；King Mongkut’s University of Technology Thonburi (KMUTT)（蒙固皇家理工大学）
For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda and epsilon >0 approximates the partition function within a relative error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f from above and N is the number of monomials in f. As a corollary, we obtain a quasi-polynomial algorithm, which, given such an f with coe…
A $\{00,01,10,11\}$-valued function on the vertices of the $n$-cube is called a $t$-resilient $(n,2)$-function if it has the same number of $00$s, $01$s, $10$s and $11$s among the vertices of every subcube of dimension $t$. The Friedman and Fon-Der-Flaass bounds on the correlation immunity order say that such a function must satisfy $t\le 2n/3-1$; moreover, the $(2n/3-1)$-resilient $(n,2)$-functions correspond to the equitable partitions of the $n$-cube with the quotient matrix $[[0,r,r,r],[r,0,r,r],[r,r,0,r],[r,r,r,0]]$, $r=n/3$. We suggest constructions of such functions and corresponding pa…
We consider the problem of computing the partition function $\sum_x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic polynomial $f$, we show that the partition function can be approximated within relative error $0 < ε< 1$ in quasi-polynomial $n^{O(\ln n - \ln ε)}$ time if the Lipschitz constant of the non-linear part of $f$ with respect to the $\ell^1$ metric on the Boolean cube does not exceed $1-δ$, for any $δ>0$, fixed in advance. For a cubic polynomial $f$, we get the same result unde…
Assuming that a plane partition of the positive integer $n$ is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as $n$ becomes large. The generating functions that arise in this study are of the form $Q(x)F(x)$, where $Q(x)=\prod_{j=1}^\infty (1-x^j)^{-j}$ is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function $F(x)$ around $x=1$. The representat…
