Simone Severini
We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M=AB, where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D.
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A solution to Babai's problem on digraphs with non-diagonalizable adjacency matrix
本文通过分别构造无限族$s$-弧传递有向图和点本原有向图，解决了Babai关于寻找邻接矩阵不可对角化的有向图的开放问题。
Natural join of $di-bigraphs$ that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their $Hasse$ digraphs called $KoDAGs$. $KoDAGs$ are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. $KoDAGs$ appear to be distinguished family of $Ferrers$ digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called $di-bicliques$. These digraphs serve to represent faithfully correspon…
For a digraph D and three parameters x, y, z in {0,1,+,-} we define the digraph D^(x,y,z) and call it the (x,y,z)-transformation of D. We show that for every r-regular digraph D the adjacency characteristic polynomial A(t, D^(x,y,z)) of (x,y,z)-transformation of D is uniquely defined by r and the adjacency characteristic polynomial A(t, D) of digraph D and we give a description of this function A(t, D^(x,y,z)) = F(r, A(t, D)). We also obtain similar results for some non-regular digraphs, namely, for so-called digraph-functions and their inverse. Also using the (x,y,z)-transformations of digrap…
The paper gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from $x$ to $y$ is equal to the complex unity $i$ (and its symmetric entry is $-i$) if the reverse arc $yx$ is not present. We also allow arcs in both directions and unoriented edges, in which case we use $1$ as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced su…
The spectra of digraphs with Morita equivalent $C^\ast$-algebras
本文研究保持图 $C^\ast$-代数 Morita 等价的有向图移动是否保持各类有向图谱，明确 adjacency 等谱的保持移动并用反例否定其余情形。
We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give tables by computer on the identification of digraphs by their eigenvalues.
In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory.
On the Dual Drazin Inverse of Adjacency Matrices of Dual-number-Weighted Digraphs
本文研究了双数权重有向图邻接矩阵的双Drazin逆，推导了相关显式公式并应用于不同类型的有向图。
A normally regular digraph with parameters $(v,k,λ,μ)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+λ(A+A^t)+μ(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $μ$ common out-neighbours, a pair of vertices connected by an edge in one direction have $λ$ common out-neighbours and a pair of vertices connected by edges in both directions have $2λ-μ$ common out-neighbours. We often assume that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regu…
Fix $c\in (0,1)$ and let $Γ$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $Γ$ is invertible with high probability. The proof uses a couplings approach based on the switchings method of McKay and Wormald. We also rely on discrepancy properties for the distribution of edges in $Γ$, recently proved by the author, to overcome certain difficulties stemming from the dependencies between the entries of $M$.
Every nonsymmetric $4$-class association scheme can be generated by a digraph
该研究在特定假设下刻画了恰含一对非对称关系的交换结合方案的有向图生成条件，并应用该结论证明所有非对称4类结合方案都可由有向图生成，拓展了3类结合方案生成相关结论。
We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The proof makes use of a coupling of random regular digraphs formed by "shuffling" the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges recently obtained by the author (arXiv:1410.5595). We also apply our general approach to prove a.a.s.\ invertibility of Hadamard products $Σ\circ Ξ$, where $Ξ$ is a matrix of iid u…
On the singularity and the inverse of 3-colored digraphs
本文研究了3色有向图的非奇异性和逆矩阵特性，探讨了非奇异单环和双环有向图的特征，并分析了其逆矩阵的零对角线条件及可逆性。
We extend Hoffman's coclique bound for regular digraphs with the property that its adjacency matrix is normal, and discuss cocliques attaining the inequality. As a consequence, we characterize skew-Bush-type Hadamard matrices in terms of digraphs. We present some normal digraphs whose vertex set is decomposed into disjoint cocliques attaining the bound. The digraphs provided here are relation graphs of some nonsymmetric association schemes.
Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al. \cite{LWCL} extended the definition to digraphs. For any real $α\in[0,1]$, the matrix $A_α(G)$ of a digraph $G$ is defined as $$A_α(G)=αD(G)+(1-α)A(G).$$ The largest modulus of the eigenvalues of $A_α(G)$ is called the $A_α$ spectral radius of $G$, denoted by $λ_α(G)$. This paper proves some extremal results about the spe…
We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly $\ell$-walk-regular with $\ell >1$ if the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph $Γ$ with only real eigenvalues whose adjac…
We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix, respectively. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equ…
The idiosyncratic polynomial of a graph $G$ with adjacency matrix $A$ is the characteristic polynomial of the matrix $ A + y(J-A-I)$, where $I$ is the identity matrix and $J$ is the all-ones matrix. It follows from a theorem of Hagos (2000) combined with an earlier result of Johnson and Newman (1980) that the idiosyncratic polynomial of a graph is reconstructible from the multiset of the idiosyncratic polynomial of its vertex-deleted subgraphs. For a digraph $G$ with adjacency matrix $A$, we define its idiosyncratic polynomial as the characteristic polynomial of the matrix $ A + y(J-A-I)+zA^{T…
The traditional adjacency matrix of a mixed graph is not symmetric in general, hence its eigenvalues may be not real. To overcome this obstacle, several authors have recently defined and studied various Hermitian adjacency matrices of digraphs or mixed graphs. In this work we unify previous work and offer a new perspective on the subject by introducing the concept of monographs. Moreover, we consider questions of cospectrality.
On the Adjacency spectra of alternating-oriented $n$-gonal staircase digraphs
研究了交替定向n边阶梯有向图的邻接谱，通过分析其邻接矩阵的结构和性质，确定了非零特征值的分布和重数，并与Padovan螺旋数相关联。
A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the distance matrix, distance Laplacian, and distance signless Laplacian, in addition to the adjacency matrix, Laplacian, and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method…
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $δ_i$ be the i…
Spectrally symmetric orientations of graphs
本文研究了非二分图的谱对称定向图，探讨了其密度极值问题及构造方法。
On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$
本文构建了无限序列的强正则有向图，通过将邻接矩阵表示为由循环块组成的块矩阵，并结合多项式模$x^{2n+3}-1$的紧缩操作，发现稳定的结构模式，从而推导出邻接矩阵的显式公式，并给出了两个此前未解决的图参数例子。
The way to arrive at formula of zeta matrix for any graded posets with the finite set of minimal elements is delivered following the first reference. This is being achieved via adjacency and zeta matrix description of bipartite digraphs chains, the representatives of graded posets. The bipartite digraphs elements of such chains amalgamate to form corresponding cover relation graded poset digraphs with corresponding adjacency matrices being amalgamated throughout natural join as special adequate database operation. The colligation of reachability and connectivity with the presented description …
A weakly distance-regular digraph is thick if its attached scheme is regular. In this paper, we show that each commutative thick weakly distance-regular digraph has a thick weakly distance-regular subdigraph such that the corresponding quotient digraph falls into six families of thick weakly distance-regular digraphs up to isomorphism.
Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. For any real $α\in[0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_α(G)$ as $$A_α(G)=αD(G)+(1-α)A(G).$$ The largest modulus of the eigenvalues of $A_α(G)$ is called the $A_α$ spectral radius of $G$. In this paper, we determine the digraphs which attain the maximum (or minimum) $A_α$ spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also discuss a number of ope…
Weakly distance-regular digraphs of diameter 2
本文研究距离正则图的定向版本，刻画了所有直径为2的弱距离正则有向图。
Weakly distance-regular digraphs whose underlying graphs are distance-regular,II
本文延续既有工作，分类了底图为Johnson图或折叠Johnson图的全部交换弱距离正则有向图。
A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we focus on commutative quasi-thin weakly distance-regular digraphs, and classify such digraphs with valency more than 3. As a result, this family of digraphs are completely determined.
Weakly distance-regular digraphs whose underlying graphs are distance-regular, I
本文针对距离正则图的定向何时构成弱距离正则有向图的问题，分类了底图分别为Hamming图、折叠n立方体和Doob图的交换弱距离正则有向图。
Semicomplete multipartite weakly distance-regular digraphs
本文引入双正则队际半完全多部有向图，证明其分为三类，并刻画了所有半完全多部交换弱距离正则有向图。
In this paper, we study commutative weakly distance-regular digraphs whose attached association schemes are regular, and give a characterization of mixed arcs. As an application, we classify such digraphs of diameter 2.
$P$-polynomial weakly distance-regular digraphs
本文定义了伴随方案为$P$-多项式的弱距离正则有向图，并刻画了所有此类有向图。
We classify certain non-symmetric commutative association schemes. As an application, we determine all the primitive weakly distance-regular circulant digraphs.
Suzuki (2004) [7] classified thin weakly distance-regular digraphs and pro- posed the project to classify weakly distance-regular digraphs of valency 3. The case of girth 2 was classified by the third author (2004) [9] under the assumption of the commutativity. In this paper, we continue this project and classify these digraphs with girth more than 2 and two types of arcs.
Non-Hamiltonian 2-regular Digraphs
该研究将2-正则有向图的非哈密顿性研究拓展至更大有向图类，引入路径与商概念给出更多判定准则，识别出若干非哈密顿正则有向图族，填补相关文献稀缺的空白。
In this paper, we classify commutative weakly distance-regular digraphs of valency 3 with girth more than 2 and one type of arcs. Combining [8, Theorem 1.2], [10, Theorem 1.3] and [11, Theorem 1], commutative weakly distanceregular digraphs of valency 3 are completely determined.
A weakly distance-regular digraph is 3-equivalenced if its attached association scheme is 3-equivalenced. In this paper, we classify the family of such digraphs under the assumption of the commutativity.
Locally semicomplete weakly distance-regular digraphs
本文针对局部半完全弱距离正则有向图展开研究，在交换性假设下完成了对这类有向图的全部刻画，为弱距离正则有向图的分类研究提供了相关结果。
A weakly distance-regular digraph is quasi-thin if the maximum value of its intersection numbers is 2. In this paper, we show that the valency of any commutative quasi-thin weakly distance-regular digraph is at most 6.
In this paper, we show that the edge connectivity of a distance-regular digraph $Γ$ with valency $k$ is $k$ and for $k>2$, any minimum edge cut of $Γ$ is the set of all edges going into (or coming out of) a single vertex. Moreover we show that the same result holds for strongly regular digraphs. These results extend the same known results for undirected case with quite different proofs.
Let $G$ be a digraph with adjacency matrix $A(G)$ and outdegrees diagonal matrix $D(G)$. For any real $α\in[0,1]$, the $A_α$ matrix $A_α(G)$ of a digraph $G$ is defined as $A_α(G)=αD(G)+(1-α)A(G)$. The eigenvalue of $A_α(G)$ with the largest modulus is called the $A_α$ spectral radius of $G$. In this paper, we first give some upper bounds for the $A_α$ spectral radius of a digraph and we also characterize the extremal digraphs attaining these bounds. Moreover, we define the $A_α$ energy of a digraph $G$ as $E^{A_α}(G)=\sum\limits_{i=1}^n(λ^α_i(G))^2$, where $n$ is the number of vertices and $λ…
In this paper, we classify all commutative weakly distance-regular digraphs of girth $g$ and one type of arcs under the assumption that $p_{(1,g-1),(1,g-1)}^{(2,g-2)}\geq k_{1,g-1}-2$. In consequence, we recover [13, Theorem 1.1] as a special case of our result.
Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D) = diag(D) + A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give sharp bound on q(D) with outdegree sequence and compare the bound with some known bounds, establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the corresponding extremal …
Network Routing on Regular Digraphs and Their Line Graphs
本文研究正则有向图上的全对全网络路由，证明每个连通正则有向图均存在无等待路由方案，且该方案随直径下降而更高效，同时证明“最远距离优先”算法在Kautz图等网络中能产生最优调度。
A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every vertex~$v$, the two edges directed away from $v$ are not consecutive in the local rotation around $v$. In other words, at each vertex the incident edges are oriented in-out-in-out. The goal of this article is to provide an analogue of Whitney's theorem on planar embeddings in the setting of 2-regular digraphs. In the course of doing so, we note that Tutte's Theorem on…
Detecting Graphical and Digraphical Regular Representations in groups of squarefree order
本文精确刻画了无平方因子阶群中，哪些群满足DRR-检测或GRR-检测性质，即通过自同构固定子集的条件充分性来判定正则表示的存在性。
Let $G=(V(G) ,E(G))$ be a digraph without loops and multiarcs, where $V(G)=\{v_1,v_2,\ldots,v_n\}$ and $E(G)$ are the vertex set and the arc set of $G$, respectively. Let $d_i^{+}$ be the outdegree of the vertex $v_i$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)=\textrm{diag}(d_1^{+},d_2^{+},\ldots,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. The spectral radius of $Q(G)$ is called the signless Laplacian spectral radius of $G$, denoted by $q(G)$. In this paper, some upper bounds for $q(G)$ a…
Localization and metric dimension for families of highly structured digraphs
研究强正则图及其推广的有向类似图族的度量维数和定位博弈，采用概率方法，得出定位数和度量维数上界，给出特定有向图族的相关结果。
In this paper, we completely characterize the digraphs of order $n$ whose $m$-step competition graphs are star graphs for positive integers $2\leq m < n$. This result in matrix version identifies the solution set to the matrix equation $X^m(X^T)^m= Λ_n+I_n$ for positive integers $2\leq m < n$ where $I_n$ is the identity matrix of order $n$ and $Λ_n$ is a $(0,1)$ Boolean matrix such that the first row and the first column consist of $1$'s except $(1,1)$-entry and the remaining entries are $0$, which is the adjacency matrix of a star graph of order $n$. We also derive meaningful properties of th…
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.
It is known that every distance-regular digraph is connected and normal. An interesting question is: when is a given connected normal digraph distance-regular? Motivated by this question first we give some characterizations of weakly distance-regular digraphs. Specially we show that whether a given connected digraph to be weakly distance-regular only depends on the equality for two invariants. Then we show that a connected normal digraph $\G$ with $d+1$ distinct eigenvalues is distance-regular if and only if the simple excess (the ratio of the square of mean of the numbers of shortest paths be…
The support of a matrix M is the (0,1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M. This paper observes some structural properties of digraphs of unitary matrices.
In this paper, we consider upper bounds on the size of transitive subtournaments in a digraph. In particular, we give an analogy of Hoffman's bound for the size of cocliques in a regular graph. Furthermore, we partially improve the Hoffman type bound for doubly regular tournaments by using the technique of Greaves and Soicher for strongly regular graphs [4], which gives a new application of block intersection polynomials.
We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint Hamilton cycles. Here G is a robust outexpander if for every set S which is not too small and not too large, the `robust' outneighbourhood of S is a little larger than S. This generalises a result of Kühn, Osthus and Treglown on approximate Hamilton decompositions of dense regular oriented graphs. It also generalises a result of Frieze and Krivelevich on approximate Hamilt…
On the existence of directed strongly regular graphs with parameters (22, 9, 6, 3, 4)
本文构造了一族参数为 (22, 9, 6, 3, 4) 的有向强正则图，其邻接矩阵由 3×3 循环块构成，自同构群为 Z_3，并用骨架与索具概念刻画其结构。
Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geqλn$. We show that for any $κ\geq 0$, \[\mathbb{P}[s_n(A)\leqκ]\leq C_λκ\sqrt{n}+2e^{-c_λn}.\] Up to the constants $C_λ, c_λ> 0$, our bound matches optimal bounds for $n\times n$ random matrices, each of whose entries is an i.i.d $\text{Ber}(d/n)$ random variable. The special case $κ= 0$ of our result confirms a conjecture of Cook regarding the probability of singularity of dense random regular digraphs.
Zero-Sum Cycles in Regular Digraphs
本文研究正则有向图中的零和环，证明了零和环存在的阈值，给出了顶点不交与边不交零和环的数量下界，结果渐近最优，相关结论可推广至欧拉有向图，所用技术基于Friedland的行列式-永久式论证。
New Bounds for the Spectral Radius and Low Energy of the $A_α$-Matrix of Digraphs
本文研究了有向图A_α-矩阵的谱半径和低能量的新界，提出了两种Koolen-Moulton型上界，并通过数值比较验证了其有效性，同时恢复了经典结果。
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