Denis Krotov
We derive recursive and direct formulas for the interweight distribution of an equitable partition of a hypercube. The formulas involve a three-variable generalization of the Krawtchouk polynomials. Keywords: equitable partition; regular partition; partition design; strong distance invariance; interweight distribution; distance distribution; Krawtchouk polynomial
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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.
Equivalence relations on the edge set of a graph $G$ that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of $G$, which in turn give rise to quotient graphs that can have a rich product structure even if $G$ itself is prime.
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.
We finish the classification of equitable 2-partitions of the Johnson graphs of diameter 3, $J(n,3)$, for $n>10$.
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.
In the monograph 'Association schemes', C. Godsil derived a necessary condition for equitable partitions of association schemes and noticed that it could be used to show that certain equitable partitions do not exist. In this short note, we show that, in fact, this condition is not stronger than the well-known Lloyd theorem.
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.
Equitable [[2,10],[6,6]]-partitions of the 12-cube
本文利用计算机辅助完成了12-方体具有商矩阵[[2,10],[6,6]]的均衡划分的分类，发现103个等价类及40个不相交简单正交阵列对的等价类，并探讨了非简单正交阵列的存在性。
In this paper we introduce a construction of directed strongly regular graphs from smaller ones using equitable partitions. Each equitable partition of a single DSRG satisfying several conditions leads to an infinite family of directed strongly regular graphs. We construct in this way dozens of infinite families. For order at most 110, we confirm the existence of DSRGs for 30 previously open parameter sets.
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 consider constructions of covering-radius-1 completely regular codes, or, equivalently, equitable 2-partitions (regular 2-partitions, perfect 2-colorings), of halved n-cubes. Keywords: completely regular code, equitable partition, regular partition, partition design, perfect coloring, halved hypercube.
Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs
本文提出有向图中分支公平划分的两个扩展：混合图中匹配森林的单准则公平划分和数字有向图中 $b$-分支的公平划分，并证明存在性及多面体的整数分解性质。
An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned into $k-1$ induced forests. We also prove that for any integers $d\ge 1$ and $k\ge 3^{d-1}$, any $d$-degenerate graph can be equitably partitioned into $k$ induced forests. Each of these results implies the existence of a constant $c$ such that for any $k \ge c$, any planar graph has an equitable partition into $k$ induced forests. This was conjectured by W…
The Kuramoto model describes the collective dynamics of a system of coupled oscillators. An alpha-Kuramoto partition is a graph partition induced by the Kuramoto model, when the oscillators include a phase frustration parameter. We prove that every equitable partition is an alpha-Kuramoto partition, but that the converse does necessarily not hold. We give an exact characterisation of alpha-Kuramoto bipartitions.
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
本文研究了正则图的公平划分条件，并基于此推导了正常赛尔图中存在(a,b)-完美集的必要条件，涉及图的不可约特征标。
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…
An equitable partition for the distance-regular graph of the bilinear forms
本文研究双线性形式图的距离正则图的公平划分，提出(x,y)-划分，并将其特征向量作为T-模块的基础，分解为不可约T-模块的正交直和。
Spectral properties of random graphs with fixed equitable partition
本文定义$S$-正则图及其矩阵，推导大型随机$S$-正则矩阵的极限谱密度和特征向量统计量，并证明扩张器混合引理、Alon-Bopana界等特征值不等式。
From semi-total to equitable total colorings, and their induced partitions into perfect and total perfect codes
研究通过改进的Kempe算法获取对称立方图和笼图的公平总着色，探讨总着色猜想的可行性。
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…
Quotient graphs and stochastic matrices
本文研究了具有相同商图的图之间的关系，证明了具有相同对称商的图与双随机矩阵存在性相关的结构性结果，并推导出具有相同组合商的新刻画，同时探讨了带权顶点图与伪公平划分，结果适用于量子游走研究。
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 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…
We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where $t_{G}\Spec(G) \subseteq \ZZπ$, and $H$ is a circulant with odd eigenvalues, their weak product $G \times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a g…
We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depen…
A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance two must receive distinct integers. The lambda chromatic number (or the $λ$ number) of a graph $G$ is the least positive integer among all the maximum assigned positive integer over all possible lambda colouring of the graph $G$. Here we have primarily shown that every graph with lambda chromatic number $t$ can be embedded in a graph, with lambda chromatic n…
Quantum fractional revival on zero-divisor graphs over $\mathbb{Z}_n$
研究$\mathbb{Z}_n$零因子图上连续时间量子行走中完美态转移和分数复兴的存在条件，通过等价划分确定顶点对的充分条件，并分析分数复兴在等价划分中的限制。
We list all perfect colorings of $Z^2$ by 9 or less colors. Keywords: perfect colorings, equitable partitions
The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n,k). This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine the eigenvalues of A(n,k) for small values of k. Finally, we show that any eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n…
Minimal representations of topology-preserving quantum-like states
本文提出公平划分方法，为量子类比特的图笛卡尔积构造保留谱与拓扑性质的最小表示，并给出直接构造及拓扑纤维化解释。
On unbalanced Boolean functions with best correlation immunity
研究达到相关免疫性界限的非平衡布尔函数，证明了1和0数量乘积可被3整除，确定了12立方体等分划分的等价类，并表征了相应的正交阵列。
Cluster Synchronization via Graph Laplacian Eigenvectors
本文提出基于图拉普拉斯特征向量的谱框架，形式化几乎等分划分与簇同步的联系，并引入准等分划分以处理现实网络中的不规则性，从而弥合静态拓扑与动态行为之间的鸿沟。
On combinatorial structure and algebraic characterizations of distance-regular digraphs
本文研究距离正则有向图，基于等度划分揭示其组合结构，并据此提出包括谱盈余定理在内的新代数刻画，同时重新推导出已有刻画。
We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $Γ$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $Γ$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$. We partition $X$ according to the path-length distance to both $x$ and $C$. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a $\mathbb{C}$-vector space ${\bf W}$. The universal d…
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