Octavio Arizmendi, Jorge Fernandez Hidalgo
We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through elementary methods.
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Strict Partitions of Maximal Projective Degree
对于所有n < 101的严格划分，计算其投影度，并找出每个n的具有最大投影度的划分。据观察，在某种意义上，最大化n的连续值的分区“彼此接近”。假设这适用于较大的n值，则计算所有n < 221的最大度的划分。这一结果与最近关于最大投射度严格划分的极限形状的一个猜想是一致的。
Sharp bounds for the number of maximal independent sets in trees of fixed diameter
我们得到了固定顶点数和直径的树中极大（包含下）独立集个数的精确下界和上界。所有的极值树被描述为同构。
In 1991, Bollobás and Frieze showed that the threshold for $G_{n,p}$ to contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar subgraph of fixed girth $g$.
On alternating signed permutations with the maximal number of fixed points
由R. Stanley关于一类交错置换的研究，R. Chapman和L.威廉姆斯指出，交替排列的最大数量的不动点是均匀分布的无序。我们扩展这个（类型$A$）的结果，类型$B$：我们证明了各类交替签署置换的最大数量的不动点是等分布的某些类型的错位（类型$B$），分别。
Maximal degree subposets of $ν$-Tamari lattices
本文研究ν-Tamari格中含最大入/出度元素的两个子偏序集，证明其最大度为弱拟合ν的最大阶梯形路径大小，且最大出/入度偏序集分别与(m-1)-Dyck路径的ν-Tamari格、加贪心序的(m-1)-Dyck路径偏序同构，还给出ν-Tamari格性质。
Maximal Algebraic Connectivity for Paths with Fixed Effective Resistance
该研究探讨固定总有效电阻下最大化图代数连通度的边权值问题，提出最大值由自同构不变权值取得的猜想，并给出$P_3$、$P_4$路径的显式解验证猜想成立。
For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erdös, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum number of maximal independent sets in a graph on n vertices, as well as the extremal graphs. In this paper we maximize the number of maximal independent sets of a fixed size for all graphs of order n and determine the extremal graphs. Our result generalizes the classical result.
Graphs with Given Degree Sequence and Maximal Spectral Radius
本文描述了在给定度序列的所有连通图中具有最大谱半径的图的结构。我们表明，在这样的图的度序列是非增加的宽度优先搜索引起的顶点的顺序。对于树的结构是唯一确定的同构。我们还表明，最大的谱半径在这样的类树是严格单调的优化。
We prove that every graph of minimum degree at least $d \ge 1$ contains a subdivision of some maximal 3-degenerate graph of order $d+1$. This generalizes the classic results of Dirac ($d=3$) and Pelikán ($d=4$). We conjecture that for any planar maximal 3-degenerate graph $H$ of order $d+1$ and any graph $G$ of minimum degree at least $d$, $G$ contains a subdivision of $H$. We verify this in the case $H$ is $P_6^3$ and $P_7^3$
Bijections on rooted trees with fixed size of maximal decreasing subtrees
Seo和Shin证明了$[n+1]$上的有根树的数目使得具有相同根的最大递减子树具有$k+1$顶点等于函数$f:[n]\to[n]$的数目使得$f$的图像包含$[k]$。我们给出了这个定理的一个双射证明。
On the enumeration of rooted trees with fixed size of maximal decreasing trees
令$\T_{n}$是$\set{0,...,n}$上的有根标记树的集合。根标号树的最大递减子树定义为从根开始的所有边都递减的最大子树。本文研究了$\T_n$的一个新的加细$\T_{n,k}$，它是最大递减子树有$k+1$顶点的有根标号树的集合.
Maximal cross-ratio degree for 8 points in $\mathbb{P}^1$
本文证明了在P¹上8点的最大交叉比次数为4。
Given a graph $G$, viewed as a loop-less symmetric digraph, we study the maximum number of fixed points in a conjunctive boolean network with $G$ as interaction graph. We prove that if $G$ has no induced $C_4$, then this quantity equals both the number of maximal independent sets in $G$ and the maximum number of maximal independent sets among all the graphs obtained from $G$ by contracting some edges. We also prove that, in the general case, it is coNP-hard to decide if one of these equalities holds, even if $G$ has a unique induced $C_4$.
On the ratio of maximum and minimum degree in maximal intersecting families
为了研究一个最大相交族$\mathcal{F}\subseteq \binom{[n]}{r}$的平衡或不平衡程度，我们考虑它的最大度和最小度之比$\mathcal{R}(\mathcal{F})=\frac{Δ(\mathcal{F})}{δ(\mathcal{F})}$。我们确定了函数$m(n,r)$的数量级，即$\mathcal{R}(\mathcal{F})$的最小可能值，并建立了函数$M(n,r)$的一些上下上界，即$\mathcal{R}(\mathcal{F})$的最大可能值。为了获得结构，显示的界限$m(n,r)$，我们使用一个定理Blokhuis的最小尺寸的非平凡块集在射影平面。
In this paper, we study the maximum degree in non-empty induced subgraphs of the Kneser graph $KG(n,k)$. One of the main results asserts that, for $k>k_0$ and $n>64k^2$, whenever a non-empty subgraph has $m\ge k{n-2\choose k-2}$ vertices, its maximum degree is at least $\frac 12(1-\frac {k^2}n) m - {n-2\choose k-2}\ge 0.49 m$. This bound is essentially best possible. One of the intermediate steps is to obtain structural results on non-empty subgraphs with small maximum degree.
A geometric approach to Mathon maximal arcs
丹尼斯顿在1969年给出了一个建设最大的弧度d在Deskiesian射影平面的偶数阶q，为所有d除q。2002年，Mathon给出了一种构造方法，推广了Denniston的构造方法。我们将给出一个新的几何方法，这些极大弧。这将允许我们计算PG（2，2^h），h素数中8度Mathon极大弧的同构类的数量。
On the maximal energy tree with two maximum degree vertices
对于一个简单图$G$，能量$E(G)$被定义为它的相邻矩阵的所有特征值的绝对值之和。对于$Δ\geq 3$和$t\geq 3$，用$T_a(Δ,t)$（或简称$T_a$）表示通过在路径$P_t$的每一端上附加$Δ-1$ $P_2$而由$t$顶点上的路径$P_t$形成的树，以及用$T_b(Δ, t)$（或简称$T_b$）表示通过在$P_{t+2}$的一端上附加$Δ-1$ $P_2$和在紧邻该端的顶点上附加$Δ-2$ $P_2$而由$P_{t+2}$形成的树.在[X. Li，X. Yao，J. Zhang和I。Gutman，Maximum energy trees with two maximum degree vertices，J.Math.Chem.45（2009），962- 973]，Li等人证明了在具有两个最大度顶点$Δ$的$n$阶树中，最大能量树是图$T_a$或图$T_b$，其中$t=n+4-4Δ\geq 3$。然而，他们不能确定$T_a$和$T_b$中的哪一个是最大能量树。这是因为准序方法对于比较它们的能量是无效的。本文利用一种新的方法来确定最大能量树。事实证明，事情更加复杂。我们证明了对于$Δ\geq 7$和任何$t\geq 3$，最大能量树都是$T_b$，而对于$Δ=3$和任何$t\geq 3$，最大能量树都是$T_a$.此外，对于$Δ=4$，除了$t=4$之外的所有$…
We study the maximal number of pairwise distinct columns in a $Δ$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $Δ$. We complement this and obtain an upper bound of the form $O(Δ)$ for fixed $m$, and with the implied constant depending polynomially on $m$.
It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the equality, i.e., the lattice simplices of dimension $(4s-2)$ with degree $s$ which are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as "Cayley conjecture". Moreover, by modifying Nill's inequaitly sli…
On Mathon's construction of maximal arcs in Desarguesian planes. II
Mathon在最近的一篇文章[M]中给出了极大弧的一个新的构造，它推广了Denniston的构造.关于这个建设，马松问的问题，确定最大程度的非丹尼斯顿最大弧所产生的新建设。在本文中，我们给这个问题一个近乎完整的答案。特别地，我们证明了当$m\geq 5$和$m\neq 9$，由{p，1}-映射生成的PG（2，2 ^m）中的非Denniston极大度弧$2^d$的最大$d$是$(\floor {m/2} +1)$ .这证实了我们在[FLX]中的猜想。对于{p，q}-映射，证明了如果$m\geq 7$和$m\neq 9$，则由{p，q}-映射生成的PG（2，2 ^m）中非Denniston极大度弧$2^d$的最大$d$是$\floor {m/2} +1$或$\floor{m/2} +2$ .
To estimate the lower bound for the chromatic number of a graph $G$, Lovász associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood complex and relates the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent ei…
Turán's problem for trees $T_n$ with maximal degree $n-4$
对于$n\ge 6$，设$V=\{v_0,v_1,\ldots,v_{n-1}\}$，$E_1=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2}$，$v_1v_{n-1}\}$，$E_2=\{v_0v_1,\ldots,v_0v_{n-4},v_1v_{n-3},v_1v_{n-2},v_2v_{n-1}\}$，$E_3=\{v_0v_1,\ldots,v_0v_{n-4}$，$v_1v_{n-3},v_2v_{n-2},v_3v_{n-1}\}$，$T_n^3=(V,E_1),\ T_n^{''}=(V,E_2)$和$T_n^{'''} =(V,E_3).$本文给出了$p\ge n\ge 15$的$ex(p;T_n^3)$，$ex(p;T_n^{''})$和$ex(p;T_n^{'''})$的显式公式，其中$ex(p;L)$表示不含$L$作为子图的$p$阶图的最大边数。
Let $G$ be a connected graph with minimum degree $δ(G)$ and vertex-connectivity $κ(G)$. The graph $G$ is $k$-connected if $κ(G)\geq k$, maximally connected if $κ(G) = δ(G)$, and super-connected (or super-$κ$) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is $k$-connected, maximally connected or super-connected if the number of edges or the spectral radius is large enough.
Maximal Degree in the Strong Bruhat Order of Bn
给定Sn中的置换P，设G（P）是n个顶点{1，.，n}，其中两个顶点i<j是相邻的，如果i出现在P中j的右边，并且P中不存在i<k<j且k出现在i和j之间的整数k.设G '（P）是通过丢弃i出现在j的右边的条件而得到的图，即如果矩形[i，P（i）] x [j，P（j）]是空的，则两个顶点是相邻的.在研究置换上的强序时，Adin和Roichman引入了这类图，并计算了它们的最大边数。我们将这些结果推广到符号置换Bn的外尔群，处理顶点{-n，.，n}\{0}，使用图兰经典定理的新变体。
Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. It is known that for a fixed $n$ one has \[ c_n r^n < m(n,r) < C_n r^n. \] We prove that for any fixed $n$ the sequence $a_r := m(n,r)/r^n$ has a limit, which was conjectured by Alon. We also prove the list colorings analogue of this statement.
Digraphs with a fixed number of edges and vertices, having a maximal number of walks of length 2
灵感的工作Backelin的非交换对应麦考利定理的增长希尔伯特系列的仿射代数，我们研究嵌入维依赖版本，他的程度2度3的结果。在图论术语中，我们研究了以下问题：在一个有（k）条边和（n）个顶点的有向图中，长度为2的有向游动的最大数目是多少？这个问题也可以用公式表示如下：当（λ）是（k）的一个分割，包含在（n \times n）盒中时，最大化（< λ，λ^T >）。我们表明，对于温和的限制（n），最佳有向图是“饱和星的明星”。
Partial flocks of the quadratic cone yielding Mathon maximal arcs
N.汉密尔顿和J. A. Thas描述了Mathon型的极大弧和二次锥的部分群之间的联系。这种联系具有相当代数的性质。在本文中，我们建立了这两个结构之间的几何连接。我们还定义了一个组合物上的羊群飞机，并使用这个工作出一个类似的合成版本的Mathon定理。最后，我们展示了如何有可能构建一个最大的弧Mathon型的程度2D，包含Denniston弧的程度d提供了一个解决方案，以一定的给定系统的跟踪条件。
Let $G$ be the symmetric group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that both functions $σ(G)$ and $ω(G)$ are asymptotically equal to $\frac{1}{2} \binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn, implies that $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we give a lower bound of $(1-o(1))n$ on $ω(G)$ which is independent of the classification of finite…
Maximal 2-dimensional binary words of bounded degree
该研究针对二元字母表上每个1最多邻接d个1的h×w二维字，推导了其中1的最大数量的精确公式，并借此得到矩形内最大蛇形多联骨牌长度的上界。
The general spectral radius of a graph $G$, denoted by $Θ(G,α)$, is the maximal eigenvalue of $M_α(G)=A(G)+αD(G)$ $(α\geq 0)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. A graph $G$ is called $Θ_α$-maximal in a class of connected simple graphs $\mathcal {G}$ if $Θ(G,α)$ is maximal among all graphs of $\mathcal {G}$. A $t$-cone $c$-cyclic graph is the join of a complete graph $K_t$ and a $c$-cyclic connected simple graph. Let $π=\big(d_1,d_2,\ldots,d_n\big)$ and $π'=\big(d'_1,d'_2,\ldots,d'_n\big)$ be two non-increasing degre…
A complete characterization of maximally σ-irregular trees with prescribed maximum degree
本文刻画了n≥Δ(Δ-1)+1时所有Δ≥6的极大σ-不规则树，引入三类树推导其σ-不规则性公式，证明极大树对应达到最大σ-不规则性的类，且Δ≥7时T'''类树的σ-不规则性更大。
In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $\arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.
The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.
Bifix codes and Sturmian words
本文证明了关于双缀码、episturmian字和自由群子群之间关系的一些新结果。我们研究了阶乘词集合中的bifix码。我们将一般极大bifix码的大部分性质推广到在递归字集$F$上的极大bifix码（$F$ -极大bifix码）.在Sturmian字集中含有bifix码的情况下，我们得到了几个新的结果。设$F$是一个Sturmian词集，定义为一个严格episturmian词的因子集。我们的结果表明，一个$F$ -最大bifix码的程度$d$的行为就像一组字的长度$d$的$F$。$F$ -在具有$k$字母的字母表上的Sturmian词集合中的程度$d$的最大bifix码具有$(k-1)d+1$元素。这推广了Sturmian集包含长度为$d$的$(k-1)d+1$字的事实。此外，给定一个无限字$x$，如果存在次数为$d$的有限最大双固定码$X$，使得$x$在$X$中至多具有长度为$d$的$d$因子，则$x$最终是周期的。我们的主要结果是：字母表$A$上的任何$F$ -极大度为$d$的双固定码都是~ $A$上自由群的指数为$d$的子群的基.
In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements, free arrangements and restriction counts. For graphic arrangements we can use these results to find a lower bound for the maximal degree generator in terms of triangles in the associated graph. We also apply these results to the case of hypersolvable arrangements where we define hyperexponents and use them to find a lower bound for their maximal degree generat…
The Class Edge-Reconstruction Number of a Maximal Planar Graph Is One or Two
本文确定极大平面图的类边重构数界为1或2，通过分析可翻转边删除后的补全方式等论证，解决了2010年重构数综述提出的问题。
An obstruction to the maximal singularity of the Hilbert scheme of points on threefolds
该研究对一大类次数证明三维点希尔伯特概型极大奇异性的必要条件猜想，推广了相关结果并为该领域提供了新的切空间维数分析方法。
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