Xiao-Xin Li
In this paper, the extremal graph with the maximum RRD-value among all the graphs of order n with given cut vertices or edges is characterized. In addition, an upper bounds on the reciprocal degree distance in terms of the number of cut edges is provided.
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The reciprocal degree resistance distance index of a connected graph $G$ is defined as $RDR(G)=\sum\limits_{\{u,v\}\subseteq V(G)}\frac {d_G(u)+d_G(v)}{r_G(u,v)}$, where $r_G(u,v)$ is the resistance distance between vertices $u$ and $v$ in $G$. Let $\mathscr {U}_n$ denote the set of unicyclic graphs with $n$ vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in $\mathscr {U}_n$, and characterize the corresponding extremal graph.
The graph having the minimum reduced reciprocal Randić index is characterized among the class of all unicyclic graphs with fixed number of vertices.
The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric matrix, we denote its eigenvalues as $λ_1(RD^L(G))\geq λ_2(RD^L(G))\geq \dots \geq λ_n(RD^L(G))$. The largest eigenvalue $λ_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of $n$ as a reciprocal distance Laplacian eigenvalue of $RD^L(G)$ is exactly one less than the number…
On vertex peripherians and Wiener index of graphs with fixed number of cut vertices
该研究聚焦固定割点数的图的顶点距离与Wiener指数问题，构造了含最大距离顶点的n顶点固定割点数图，并刻画了至多3个割点的n顶点图中Wiener指数最大的图的特征。
The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.
In this paper, first we compute the energy of a special partitioned matrix under some cases. As a consequence, we obtain the reciprocal distance energy of the complete multipartite graph and also we give various other energies of complete multipartite graphs. Next, we show that among all complete k-partite graphs on n vertices, the complete split graph has minimum reciprocal distance energy and the reciprocal distance energy is maximum for the Turan graph. At last, it is shown that the reciprocal distance energy of the complete bipartite graph km,m decreases under deletion of an edge if 2\le m…
The first Zagreb index of a graph $G$ is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges. In our work, we solve an open question about Zagreb indices of graphs with given number of cut vertices. The sharp lower bounds are obtained for these indices of graphs in $\mathbb{V}_{n,k}$, where $\mathbb{V}_{n, k}$ denotes the set of all $n$-vertex graphs with $k$ cut vertices and at least one cycle. As consequences, those graphs with the smallest Zagreb indices are characterized.
Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph $G$. In this work, we investigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodes of which $c$ are cut vertices. For all valid $n,c$, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of $n,c$, however, there is a unique maximal unicyclic graph with $n$ …
The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. We denote the set of all connected graphs on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote the set of all connected graphs on $n$ vertices with $s$ cut vertices by $\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper bounds on the total eccentricity index over $\mathfrak{C_{n,s}}…
A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.
A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the correspon…
The cut polytope of a graph is an important object in several fields, such as functional analysis, combinatorial optimization, and probability. For example, Sturmfels and Sullivant showed that the toric ideals of cut polytopes are useful in algebraic statistics. In the theory of lattice polytopes, the $h^{*}$-polynomial is one of the most important invariants. The necessary and sufficient condition in terms of graphs that the $h^{*}$-polynomial of a cut polytope is palindromic is known. However, except for trees, there are no classes of graphs for which the $h^{*}$-polynomial of their cut poly…
In a 2018 paper, Cameron and Semeraro posed the problem of finding all group-graph reciprocal pairs. In this paper, we make a significant contribution to finding all such pairs. A group and graph form a reciprocal pair if they satisfy the relation $$P_{Γ,G}(x)=(-1)^nF_G(-x)$$ where $P_{Γ,G}(x)$ is the orbital chromatic polynomial of a graph $Γ$ and $F_G(x)$ is the cycle polynomial of a finite permutation group. We define a set of graphs to be \textit{$k$-stars} and prove that they satisfy a reciprocality relation with some group depending on $k$. These graphs are comprised of a complete graph …
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pend…
Given a set $\Sg$ of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to $\Sg$ a canonical $X$-labeled graph $\B(\Sg)$, called the Buneman graph. This graph has several interesting mathematical properties - for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In this paper, we present some results concerning the {\em cut vertices} of $\B(\Sg)$, i.e., vertices whose removal disconnect the graph, as well as its {\em blocks} or 2-{\em connected co…
Let $G$ be a connected hypergraph with even uniformity, which contains cut vertices. Then $G$ is the coalescence of two nontrivial connected sub-hypergraphs (called branches) at a cut vertex. Let $\mathcal{A}(G)$ be the adjacency tensor of $G$. The least H-eigenvalue of $\mathcal{A}(G)$ refers to the least real eigenvalue of $\mathcal{A}(G)$ associated with a real eigenvector. In this paper we obtain a perturbation result on the least H-eigenvalue of $\mathcal{A}(G)$ when a branch of $G$ attached at one vertex is relocated to another vertex, and characterize the unique hypergraph whose least H…
Introduced by Gutman in 2021, the Sombor index is a novel graph-theoretic topological descriptor possessing potential applications in the modeling of thermodynamic properties of compounds. Let G^k_n be the set of all n-vertex connected graphs with k cut-vertices. In this paper, we present minimum Sombor indices of graphs in G^k_n. The corresponding extremal graphs have been characterized as well.
Revisiting Extremal Graphs Having No Stable Cutsets
本文填补了Le和Pfender关于边数为2n-3且无稳定割集的极图递归构造证明中的漏洞，完善了对Caro猜想相关极图结构的论证。
A graph $G$ is said to be $k$-$γ_{c}$-critical if the connected domination number $γ_{c}(G)$ is equal to $k$ and $γ_{c}(G + uv) < k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $ζ$ be the number of cut vertices of $G$ and let $ζ_{0}$ be the maximum number of cut vertices that can be contained in one block. For an integer $\ell \geq 0$, a graph $G$ is $\ell$-factor critical if $G - S$ has a perfect matching for any subset $S$ of vertices of size $\ell$. It was proved that, for $k \geq 3$, every $k$-$γ_{c}$-critical graph has at most $k - 2$ cut vertices and the graphs with max…
The cut polytope ${\rm CUT}(n)$ is the convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation with admissible correlations of symmetric Bernoulli random variables. That probabilistic interpretation is a starting point of this work in conjunction with a natural binary encoding of the CUT($n$). We show that for any $n$, with appropriate scaling, all vertices of the polytope ${\mathbf 1}$-CUT($n$) encoded as integers are approximately on the line $y= x-1…
The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n \to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.
Realizations and Uniqueness of Cut Complexes of Graphs
本文研究了图的切割复形的实现与唯一性问题，提出了参数m(d,n)并开发了O(n^4)的识别算法，揭示了图结构与拓扑之间的联系。
We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geq \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geq\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ …
Extremal graphs with minimum number of connected subgraphs in a given family
本文定义了顶点的子图数，刻画了在 $n$ 个顶点和 $k$ 个割点的所有图中具有最少连通子图数的极值图及其顶点，并在围长至少为 $k$ 的条件下得出相应结果，最终完成了对 $0 \leq k \leq 4$ 情形的完整刻画。
In this paper we study fundamental connectivity properties of hypergraphs from a graph-theoretic perspective, with the emphasis on cut edges, cut vertices, and blocks. To prepare the ground, we define various types of subhypergraphs, as well as various types of walks in a hypergraph. We then prove a number of new results involving cut edges, cut vertices, and blocks. In particular, we describe the exact relationship between the block decomposition of a hypergraph and the block decomposition of its incidence graph.
Enumeration of lozenge tilings of hexagons with cut off corners
出于对Proctor研究的一类平面划分的枚举和对平面划分的对称类的考虑，我们考虑了从六边形的一些顶点移除“最大楼梯”的菱形拼接的枚举问题。一个顶点的情况对应于Proctor问题。对于两个顶点，有几种情况需要考虑，其中大多数导致很好的枚举公式。对于三个或更多的顶点，似乎不存在一般的好的产品公式，但在一个特殊的情况下，发生了很多的因式分解，我们提出的问题，找到一个公式的数量在这种情况下的平铺。
The trace and Estrada index of uniform hypergraphs with cut vertices
本文研究含割点的一致超图邻接张量的迹，推导局部扰动下迹与Estrada指数的结果，并证明固定边数超树中超路唯一最小、超星唯一最大。
For a molecular graph, the first multiplicative Zagreb index $Π_1$ is equal to the product of the square of the degree of the vertices, while the second multiplicative Zagreb index $Π_2$ is equal to the product of the endvertex degree of each edge over all edges. Denote by $\mathbb{G}_{n,k}$ the set of graphs with $n$ vertices and $k$ cut edges. In this paper, we explore graphs in terms of a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs with given number of cut edges are provided. Furthermore, we characterize graphs with the largest and small…
On the Zagreb indices of the line graph and line cut-vertex graph of subdivision of unicyclic graphs
本文借助细分概念，研究环-星图这类单圈图的线图及线割点图的Zagreb指数与余指数。
On forest and bipartite cuts in sparse graphs
本文研究简单图中特定图类顶点割的存在条件，证明了边数少于 $(19n - 28)/8$ 的连通图存在森林割，改进了现有界，并证明边数少于 $(80n-134)/31$ 时存在二部割。
Given a finite group $G$, the character graph, denoted by $Δ(G)$, for its irreducible character degrees is a graph with vertex set $ρ(G)$ which is the set of prime numbers that divide the irreducible character degrees of $G$, and with $\{p,q\}$ being an edge if there exist a non-linear $χ\in {\rm Irr}(G)$ whose degree is divisible by $pq$. In this paper, we discuss the influences of cut vertices and eigenvalues of $Δ(G)$ on the group structure of $G$. Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of charac…
A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is connected. The connected domination number of $G$, $γ_c(G)$, is the minimum cardinality of a connected dominating set of $G$. A graph $G$ is said to be $k$-$γ_{c}$-critical if the connected domination number $γ_{c}(G)$ is equal to $k$ and $γ_{c}(G + uv) < k$ for every pair of non-adjacent vertices $u$ and $v$ of $G$. Let $ζ$ be the number of cut-vertices of $G$…
The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with maximum Harary index among all graphs with given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among the graphs with given vertex connectivity.
A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there is no isolated vertex in each component of $G-S$. The super-connectivity of graph $G$ is the size of the minimum super vertex cut of $G$. Let $KG(n,k)$ be the Kneser graph whose vertices set are the $k$-subsets of $\{1,\cdots,n\}$, where $k$ is the number of labels of each vertex in $G$. We aim to show that the conjecture from Boruzanli and Gauci \cite{EG19} …
Large odd induced subgraphs via odd cuts
研究无孤立顶点的\(n\)顶点图，利用特定二分法，得出其有至少\(n/5\)个顶点且所有度数均为奇数的导出子图，改进了该类图的通用下界。
Dinits-Karzanov-Lomonosov showed that it is possible to encode all minimal edge cuts of a graph by a tree-like structure called a cactus. We show here that minimal edge cuts separating ends of the graph rather than vertices can be `encoded' also by a cactus. We apply our methods to finite graphs as well and we show that several types of cuts can be encoded by cacti.
This paper is the first from a series of papers that provide a characterization of maximum packings of $T$-cuts in bipartite graphs. Given a connected graph, a set $T$ of an even number of vertices, and a minimum $T$-join, an edge weighting can be defined, from which distances between vertices can be defined. Furthermore, given a specified vertex called root, vertices can be classified according to their distances from the root, and this classification of vertices can be used to define a family of subgraphs called {\em distance components}. Sebö provided a theorem that revealed a relationship …
On the Automorphisms of Token Graphs Generated by $2$-cuts with the Same Neighbours
本文研究连通图G的k-令牌图自同构，证明当2-割中两顶点具有相同邻点时，可构造大量非由G的自同构诱导的令牌图自同构，并刻画了所有此类2-割生成的群。
For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prov…
The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. We study (minimal) cut-sets of the power graph of a (finite) non-cyclic (nilpotent) group which are associated with its maximal cyclic subgroups. Let $G$ be a finite non-cyclic nilpotent group whose order is divisible by at least two distinct primes. If $G$ has a Sylow subgroup which is neither cyclic nor a generalized quaternion $2$-group and all other Sylow subgroups of $G$ are cyclic, the…
A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if $G$ contains $H$ as an immersion, then for every integer $k$, the number of vertices of degree at least $k$ in $G$ is at least the number of vertices of degree at least $k$ in $H$. In this paper, we prove that this obvious necessary condition is "nearly" sufficient for graphs with no edge-cut of order 3: for every graph $H$, every $H$-immersion free graph with …
Recently Ayyaswamy [1] have introduced a novel concept of the signless Laplacian Estrada index (after here $SLEE$) associated with a graph $G$. After works, we have identified the unique graph with maximum $SLEE$ with a given parameter such as: number of cut vertices, (vertex) connectivity and edge connectivity. In this paper we continue out characterization for two further parameters; diameter and number of cut vertices.
Non-crossing frameworks with non-crossing reciprocals
我们研究平面上的非交叉框架，其对偶图上的经典倒数也是非交叉的。我们给出了一个完整的描述的自应力的非交叉框架，其倒数是非交叉的，在方面：类型的面（只允许伪三角形和伪四边形）;在自应力的符号模式;和在一些顶点的应力向量的几何条件。在其他最近的论文中，非交叉性和刚性的直线平面图的相互作用进行了研究，伪三角剖分显示为特别感兴趣的对象。例如，已知所有平面拉曼回路可以嵌入为具有一个非指向顶点的伪三角剖分。我们表明，如果这样的嵌入是足够通用的，那么倒数是非交叉，并再次伪三角嵌入平面拉曼电路。对于单数（即，非通用）平面拉曼回路的伪三角测量嵌入，倒数仍然是非交叉和伪三角测量，但其底层图可能不是拉曼回路。此外，所有的伪三角剖分，承认一个非交叉倒数出现的倒数，这种可能是奇异的，强调伪三角嵌入的拉曼电路。可平面图上的所有自应力都对应于三维空间中分段线性曲面的提升。我们证明了这种非交叉互惠对的电梯的特征几何性质。
A Short Note on Relevant Cuts
本文提出了一种基于Picard-Queyranne有向无环图的方法，用于加速相关割的枚举，并通过实验评估了其有效性。
The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert V(G)-X\rvert $ is less than $k+\ell$. We prove that every prime $3^{+2}$…
The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic increasing topological index, and characterize the extremal graphs having the minimum Wiener index, the maximum Harry index, the maximum reciprocal degree distance, the minimum eccentricity distance sum, the minimum adjacent eccentric distance sum index, the maximum connective eccentricity index, the maximum Zagreb indices among graphs with a fixed number $n…
The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. T…
Recently, Kawarabayashi and Thorup presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph $G$ on $n$ vertices whose contractions leave a multigraph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges that preserves all non-trivial min-cuts of $G$, where $δ$ is the minimum degree of $G$ and $\tilde{O}$ hides logarithmic factors. We present a simple argument that improves this contraction-based sparsifier by eliminating the poly-logarithmic factors, that is, we sho…
The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions: (1) $G$ is $k$-connected; (2) the independence number of $G$ is greater than $k;$ (3) any independent set of cardinality $k$ is a vertex cut of $G.$ The edge version of this result does not hold. We also consider the problem when replacing independent sets by the periphery.
We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n.
A graph is equimatchable if each of its matchings is a subset of a maximum matching. It is known that any 2-connected equimatchable graph is either bipartite, or factor-critical, and that these two classes are disjoint. This paper provides a description of k-connected equimatchable factor-critical graphs with respect to their k-cuts for $k\ge 3$. As our main result we prove that if G is a k-connected equimatchable factor-critical graph with at least 2k+3 vertices and a k-cut S, then G-S has exactly two components and both these components are close to being complete or complete bipartite. If b…
Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $δ(U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $Ω$ such that $w(Ω)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the…
On the Restricted Edge-Cuts of Optimal 1-Planar Graphs
本文排除最优1-平面图受限边连通度为8的可能，证明其为10或12且均可达，还明确6连通最优1-平面图受限边连通度为10的充要条件，刻画相关顶点诱导子图结构并建立受限边割与顶点割的联系。
On g-Extra Connectivity of Corona-Type Graph Products
本文研究了多种corona型图乘积的g-额外连通性，定义了R_g割集及其最小基数，并推导了边corona、邻域corona等五种图乘积的g-额外顶点和边连通性，为容错网络分析提供了理论支持。
Total cut complexes and their duals
本文研究了总割复形及其对偶体的同伦类型，解决了多个猜想并探讨了相关图结构的连通性。
This paper revisits the notion of a spanning hypertree of a hypermap introduced by one of its authors and shows that it allows to shed new light on a very diverse set of recent results. The tour of a map along one of its spanning trees used by Bernardi may be generalized to hypermaps and we show that it is equivalent to a dual tour described by Cori and Mach\`ı. We give a bijection between the spanning hypertrees of the reciprocal of the plane graph with $2$ vertices and $n$ parallel edges and the meanders of order $n$ and a bijection of the same kind between semimeanders of order $n$ and span…
We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture from the paper "Schur and e-positivity of trees and cut vertices".
Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most $d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our ar…
We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-…
We show that on every Ramanujan graph $G$, the simple random walk exhibits cutoff: when $G$ has $n$ vertices and degree $d$, the total-variation distance of the walk from the uniform distribution at time $t=\frac{d}{d-2}\log_{d-1} n + s\sqrt{\log n}$ is asymptotically $\mathbb{P}(Z > c\, s)$ where $Z$ is a standard normal variable and $c=c(d)$ is an explicit constant. Furthermore, for all $1 \leq p \leq \infty$, $d$-regular Ramanujan graphs minimize the asymptotic $L^p$-mixing time for SRW among all $d$-regular graphs. Our proof also shows that, for every vertex $x$ in $G$ as above, its distan…
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