Konstantin Vorob'ev
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.
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We finish the classification of equitable 2-partitions of the Johnson graphs of diameter 3, $J(n,3)$, for $n>10$.
Aldous property for full-flag Johnson graphs
研究证明全标志乔纳森图的谱间隙与施雷耶商相同，验证了两个猜想并揭示了其阿劳斯型谱间隙现象。
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.
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.
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.
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…
Johnson recently proved Armstrong's conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of $(b_1,b_2,\cdots, b_n)$-core partitions where $\{b_1,b_2,\cdots,b_n\}$ contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate $(s,s+1,s+2)$-core partition.
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。
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。
A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.
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
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.
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.
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。
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.
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。
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。
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。
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…
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。
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.
A $k$-coloring of a graph $G=(V,E)$ is called semi-equitable if there exists a partition of its vertex set into independent subsets $V_1,\ldots,V_k$ in such a way that $|V_1| \notin \{\lceil |V|/k\rceil, \lfloor |V|/k \rfloor\}$ and $||V_i|-|V_j|| \leq 1$ for each $i,j=2,\ldots,k$. The color class $V_1$ is called non-equitable. In this note we consider the complexity of semi-equitable $k$-coloring, $k\geq 4$, of the vertices of a cubic or subcubic graph $G$. In particular, we show that, given a $n$-vertex subcubic graph $G$ and constants $ε> 0$, $k \geq 4$, it is NP-complete to obtain a semi-e…
Motivated by an approach to visualization of high dimensional statistical data given in Hurley and Oldford (2011), this work examines the clique structure of $J_n(m, m-1)$ Johnson graphs. Cliques and maximal cliques are characterized and proved to be of one of only two types. These types are characterized by features of the intersection and of the union of the subsets of [n] = {1, 2, ..., n} which define the vertices of the graph. Clique numbers and clique partition numbers follow. The results on Johnson graphs are connected to results on intersecting families of sets related to extremal set t…
Average hitting times in some $f$-equitable graphs
本文提出 $f$-等度图这一推广距离正则图的新分类，利用商矩阵确定其中任意两顶点间的平均击中时间，并应用于两个强正则图的笛卡尔积及广义 Paley 图。
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…
Inspired by the paper [1] of V. Bergelson, John H.Johnson Jr., J. Moreira, we formulate an abstract version of image partition regularity. To establish the result we have used a variant of first entry condition and for infinite case we contained our work to Milliken-Taylor system.
The minimum number of total independent partition sets of $V \cup E$ of a graph $G=(V,E)$ is called the \emph{total chromatic number} of $G$, denoted by $χ''(G)$. If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of $V \cup E$ is called the \emph{equitable total chromatic number}, and is denoted by $χ''_=(G)$. In this paper we consider equitable total coloring of coronas of cubic graphs, $G \circ H$. It turns out that, independly on the values of equitable total chromatic number of factors $G$ a…
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…
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…
Using the theory of equitable decompositions it is possible to decompose a matrix $M$ appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, w…
We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $ϕ$, it is possible to use $ϕ$ to decompose any matrix $M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. Some of the matrices that can be decomposed are the graph's adjaceny matrix, Laplacian matrix, etc. Because this decomposition has connection…
Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(Δ(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture for graphs $G$ with $Δ(G)\geq |G|/2$.
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。
Let ${\mathcal D}_d$ be the class of $d$-degenerate graphs and let $L$ be a list assignment for a graph $G$. A colouring of $G$ such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a $d$-degenerate graph is called the $(L,{\mathcal D}_d)$-colouring of $G$. For a $k$-uniform list assignment $L$ and $d\in\mathbb{N}_0$, a graph $G$ is equitably $(L,{\mathcal D}_d)$-colorable if there is an $(L,{\mathcal D}_d)$-colouring of $G$ such that the size of any colour class does not exceed $\left\lceil|V(G)|/k\right\rceil$. An equitable $(L…
We extend the theory of equitable decompositions, in which, if a graph has a particular type of symmetry, i.e. a uniform or basic automorphism $ϕ$, it is possible to use $ϕ$ to decompose a matrix $M$ appropriately associated with the graph. The result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. We show here that a large class of automorphisms, which we refer to as \emph{separable}, can be realized as a sequence of basic automorphisms, allowing us to equitably decompose $M$ over any such automorphism. We also …
Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4- and 6-Cycles
本文证明无弦4-圈和6-圈的平面图在$k\geq\max\{\Delta(G),7\}$时是公平$k$-可着色且公平$k$-可选择的。
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…
Let $G_1$ be a planar graph such that all cycles of length at most 4 are independent and let $G_2$ be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of $G_1$ and $G_2$ can be equitably partitioned into $t$ subsets for every $t\geq 3$ so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang and Li.
We list all perfect colorings of $Z^2$ by 9 or less colors. Keywords: perfect colorings, equitable partitions
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…
Perfect colorings of the infinite square grid: coverings and twin colors
本文研究无限方格图的完美着色与覆盖问题，证明所有此类着色要么对应图自同构群作用下的轨道划分，要么具有合并后仍保持完美着色的孪生颜色，并对孪生颜色情形进行了分类。
Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are exactly (s+t-1)!/(s!t!) partitions whose set of hook-lengths is disjoint from the set {s,t}. Drew Armstrong conjectured (and Paul Johnson, and a bit later, Victor Wang, proved) a beautiful expression for the average size, namely (s-1)(t-1)(s+t+1)/24 . In the present article, we go far beyond the average, and state absolutely certain expressions (but "officially" still conjectures) for the variance (showing in particular that it is rather large, and there is no "concentration about the mean"), and the …
Preorder Construct on Simple Undirected Graphs
本文研究了简单无向图节点集上的新预序构造，证明了该预序由图拓扑保持，并通过逻辑动力系统（离散和连续时间）保持，同时该预序的等价关系对应于最粗的公平划分（CEP），为动力系统提供了非平凡不变集。
A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $χ_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In "Equitable colorings of Cartesian products of graphs" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of co…
A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the \emph{equitable chromatic number} of $G$ and it is denoted by $χ_{=}(G)$. In this paper the problem of determinig $χ_=$ for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solv…
A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $ν_u$ of the Perron eigenvector $\mathbfν$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity …
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to th…
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。
A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most $r$. The $r$-equitable chromatic threshold of a graph $G$, denoted by $χ_{r=}^*(G)$, is the minimum $k$ such that $G$ is $r$-equitably $k'$-colorable for all $k'\ge k$. Let $G\times H$ denote the Kronecker product of graphs $G$ and $H$. In this paper, we completely determine the exact value of $χ_{r=}^*(K_m\times K_n)$ for general $m,n$ and $r$. As a consequence, we show that for $r\ge 2$, if $n\ge \frac{1}{r-1}(m+r)(m+2r-1)$ then $K_m\times K_n$…
A synchrony subspace of R^n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also pres…
The $n$-Queens' graph, $\mathcal{Q}(n)$, is the graph associated to the $n \times n$ chessboard (a generalization of the classical $8 \times 8$ chessboard), with $n^2$ vertices, each one corresponding to a square of the chessboard. Two vertices of $\mathcal{Q}(n)$ are adjacent if and only if they are in the same row, in the same column or in the same diagonal of the chessboard. After a short overview on the main combinatorial properties of $\mathcal{Q}(n)$, its spectral properties are investigated. First, a lower bound on the least eigenvalue of an arbitrary graph is obtained using clique edge…
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)…
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