Maximizing the order of a regular graph of given valency and second eigenvalue

Biregular graphs with three eigenvalues

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.

关于具有4个不同特征值的共边正则图

On co-edge-regular graphs with 4 distinct eigenvalues

该研究通过构造两类具有四个不同特征值且最小特征值固定的无限族连通共边正则图，反驳了 Tan 等人的猜想，并证明 Latin Square 图的团扩张并非谱确定。

On groups all of whose undirected Cayley graphs of bounded valency are integral

A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the non-abelian case. In this paper we generalize this class of groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which all graphs $\mathrm{Cay}(G,S)$ are integral if $|S| \le k$. It will be proved that $\mathcal{G}_k$ consists of the Cayley integral groups i…

Some regular signed graphs with only two distinct eigenvalues

We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained parameter we provide some examples of signed graphs having two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of each mentioned valency with only two distinct eigenvalues. We prove that for any $k\geq 5$ there are infinitely many connected signed $k$-regular graphs having maximum eigenvalue $\sqrt{k}$. Moreover for each $m\ge…

Light tails and the Hermitian dual polar graphs

Juriśič et al. conjectured that if a distance-regular graph $Γ$ with diameter $D$ at least three has a light tail, then one of the following holds: 1.$a_1 =0$; 2.$Γ$ is an antipodal cover of diameter three; 3.$Γ$ is tight; 4.$Γ$ is the halved $2D+1$-cube; 5.$Γ$ is a Hermitian dual polar graph $^2A_{2D-1}(r)$ where $r$ is a prime power. In this note, we will consider the case when the light tail corresponds to the eigenvalue $-\frac{k}{a_1 +1}$. Our main result is: Theorem Let $Γ$ be a non-bipartite distance-regular graph with valency $k \geq 3$ , diameter $D \geq 3$ and distinct eigenvalues $θ…

On the Spectrum of the Generalised Petersen Graphs

We show that the gap between the two greatest eigenvalues of the generalised Petersen graphs $P(n,k)$ tends to zero as $n \rightarrow \infty$. Moreover, we provide explicit upper bounds on the size of this gap. It follows that these graphs have poor expansion properties for large values of $n$. We also show that a positive proportion of the eigenvalues of $P(n,k)$ tend to the valency.

最小特征值至少为$-3$的半正则图的可积性

The integrability of sesqui-regular graphs with smallest eigenvalue at least $-3$

本文证明参数满足$c\geq3$且最小特征值在$[-3,-2)$的连通半正则图在度数足够大时是$1$-可积的，并指出$c$的下界最优，通过Shrikhande图与$K_t$的笛卡尔积给出反例。

A new characterization of the dual polar graphs

In this paper we give a new characterization of the dual polar graphs, extending the work of Brouwer and Wilbrink on regular near polygons. Also as a consequence of our characterization we confirm a conjecture of the authors on non-bipartite distance-regular graphs with smallest eigenvalue at most $-k/2$, where $k$ is the valency of the distance-regular graph, in case of $c_2 \geq3$ and $a_1 =1$.

On the spectrum and linear programming bound for hypergraphs

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming method for obtaining an upper bound on the order of a regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore, we obtain a general upper bound on the order of a regular uniform…

Taut distance-regular graphs and the subconstituent algebra

We consider a bipartite distance-regular graph $G$ with diameter $D$ at least 4 and valency $k$ at least 3. We obtain upper and lower bounds for the local eigenvalues of $G$ in terms of the intersection numbers of $G$ and the eigenvalues of $G$. Fix a vertex of $G$ and let $T$ denote the corresponding subconstituent algebra. We give a detailed description of those thin irreducible $T$-modules that have endpoint 2 and dimension $D-3$. In an earlier paper the first author defined what it means for $G$ to be taut. We obtain three characterizations of the taut condition, each of which involves the…

具有固定最小特征值的有向图结构理论

A structure theory for signed graphs with fixed smallest eigenvalue

本文研究了具有固定最小特征值的有向图结构，并证明了在特定条件下，此类图具有密集子图和特定的特征值下界性质。

An inequality for regular near polygons

Let $G$ denote a near-polygon distance-regular graph with diameter $d\geq 3$, valency $k$ and intersection numbers $a_1>0$, $c_2>1$. Let $θ_1$ denote the second largest eigenvalue for the adjacency matrix of $G$. We show $θ_1$ is at most $(k-a_1-c_2)/(c_2-1)$. We show the following are equivalent: (i) Equality is attained above; (ii) $G$ is $Q$-polynomial with respect to $θ_1$; (iii) $G$ is a dual polar graph or a Hamming graph.

最小特征值至少为\(-3\)的符号图及其格

Signed graphs with fixed smallest eigenvalue at least $-3$ and their lattices

研究最小特征值至少为\(-3 - \varepsilon\)的连通符号图，证明最小度足够大时最小特征值至少为\(-3\)，相关格是标准格\(\mathbb{Z}^n\)与\(E_8\)根格直和的子格，还讨论了源于无根不可约幺模格的最小特征值为\(-3\)的符号图。

Shilla distance-regular graphs

A Shilla distance-regular graph G (say with valency k) is a distance-regular graph with diameter 3 such that its second largest eigenvalue equals to a3. We will show that a3 divides k for a Shilla distance-regular graph G, and for G we define b=b(G):=k/a3. In this paper we will show that there are finitely many Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will give a new existence condition for distance-regular graphs, in general.

有限图的第一拉普拉斯特征值最大化

Maximization of the first Laplace eigenvalue of a finite graph

本文研究有限图拉普拉斯第一非零特征值的最大化问题，证明极值解的长度函数可由第一特征函数构成的欧氏空间映射显式表示，给出Nadirashvili结果的图论类比，并得到Göring-Helmberg-Wappler问题的类似结论。

在直径约束下最大化树上的Steklov特征值

Maximizing the Steklov eigenvalues on trees with a diameter constraint

研究有限树T在直径D约束下Dirichlet到Neumann算子的第一个非零Steklov特征值λ₂(T,δΩ)，确定奇直径D=2r+1≥5的最优配置，提出广义近似锯齿树家族AS(r,q+2,c,t)。

On the Cheeger constant for distance-regular graphs

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least $\frac{λ_1}{2}$, where $λ_1$ is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for distance-regular graphs the Cheeger constant is at most $λ_1$. In particular, we prove the conjecture for the known infinite families of distance-regular graphs, distance-regular graphs of diameter 2 (the strongly regular graphs), several classes of imprimitive dista…

有限图的第一拉普拉斯特征值最大化 II

Maximization of the first Laplace eigenvalue of a finite graph II

本文研究有限图的Fujiwara拉普拉斯算子第一非零特征值的最大化问题，证明该特征值上确界有限当且仅当图为树，且该上确界在取子图时不增大。

Ramanujan circulant graphs and the conjecture of Hardy-Littlewood and Bateman-Horn

In this paper, we determine the bound of the valency of the odd circulant graph which guarantees to be a Ramanujan graph for each fixed number of vertices. In almost of the cases, the bound coincides with the trivial bound, which comes from the trivial estimate of the largest non-trivial eigenvalue of the circulant graph. As exceptional cases, the bound in fact exceeds the trivial one by two. We then prove that such exceptionals occur only in the cases where the number of vertices has at most two prime factors and is represented by a quadratic polynomial in a finite family and, moreover, under…

在正则图中最大化子图计数

Maximizing subgraph counts in regular graphs

研究正则图中最大化子图密度的问题，通过图的特征值连续优化方法，针对不同类型的图给出最优结构结论。

Two conjectures in spectral graph theory involving the linear combinations of graph eigenvalues

We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let $λ_1(G)$ be the largest eigenvalue of the adjacency matrix of a graph $G$, and $\bar{G}$ be the complement of $G$. A nice conjecture states that the graph on $n$ vertices maximizing $λ_1(G) + λ_1(\bar{G})$ is the join of a clique and an independent set, with $\lfloor n/3\rfloor$ and $\lceil 2n/3\rceil$ (also $\lceil n/3\rceil$ and $\lfloor 2n/3\rfloor$ if $n \equiv 2 \pmod{3}$) vertices, respectively. We resolve this conjecture for sufficiently large $n$ using analytic methods…

二维简单复形的谱半径与其给定贝蒂数

Maximizing the signless Laplacian spectral radius of simplicial 2-complexes with a prescribed second Betti number

本文研究了二维简单复形在给定第二贝蒂数下的无符号拉普拉斯谱半径的渐近公式及最大值复形的结构特征。

利用具有 $k$ 个悬挂顶点的生成树最大化符号完全图的指数

Maximizing the index of signed complete graphs with spanning trees on $k$ pendant vertices

本文研究负边由生成树诱导的不平衡符号完全图，确定在悬挂顶点数给定时使最大特征值最大的极值树与符号图结构。

图的最大谱间隙

Maximum spectral gaps of graphs

本文研究图的最大谱间隙问题，推广了最大展开度结果，对所有$(i,j)$给出上界，证明紧性与唯一性，推进了Nikiforov的特征值线性组合最大化研究。

论紧距离正则图的（非）存在性：一种局部方法

On the (non-)existence of tight distance-regular graphs: a local approach

本文利用距离正则图的局部结构证明紧图局部不可能是正交阵列或 Steiner 系统的区组图，并在排除这些情形后给出由参数 $b$ 控制的价与直径界。

双团与双独立对的半定近似

Semidefinite approximations for bicliques and biindependent pairs

本文研究了二部图中的双独立对参数 $g(G)$、$h(G)$ 和 $α_{\text{bal}}(G)$，证明了 $h(G)$ 及平衡最大独立集判定问题的计算困难性，并提出了基于半定规划和特征值的界，揭示了其与 Lovász $\vartheta$-数等经典谱参数的关系。

关于具有正交交数 $a_1$ 的 1-均匀距离正则图的分类

Towards a classification of $1$-homogeneous distance-regular graphs with positive intersection number $a_1$

本文研究了具有正交交数 $a_1$ 的 1-均匀距离正则图的分类，证明了在特定条件下，图要么是正则的近 $2D$-gon、Johnson 图或折叠的立方体，或其度数受函数限制。

最大化某些theta图的无符号拉普拉斯谱半径

Maximizing the signless Laplacian spectral radius of some theta graphs

本文研究三类禁用子图约束下图的无符号拉普拉斯谱半径上界，分别确定了不含θ(1,2,2)、不含θ(1,2,3)、不含{θ(1,2,2),F₅}的图的谱半径最大值及对应的极图。