Maya Stein
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.
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An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we dete…
We provide new constructions of Meyniel extremal graphs, which are families of graphs with the conjectured largest asymptotic cop number. Using spanning subgraphs, we prove that there are an exponential number of new Meyniel extremal families with specified degrees. Using a linear programming problem on hypergraphs, we explore the degrees in families that are not Meyniel extremal. We give the best-known upper bound on the cop number of vertex-transitive graphs with a prescribed degree. We find new Meyniel extremal families of regular graphs with large chromatic number, large diameter, and expl…
The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl. 609 (2021) 1-11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4. In this paper, we first determine the extremal graph which attains the maximum spect…
The family of Directed Acyclic Graphs as well as some related graphs are analyzed with respect to extremal behavior in relation with the family of intersection graphs for families of boxes with transverse intersection.
In this paper we deal with a Turán-type problem: given a positive integer n and a forbidden graph H, how many edges can there be in a graph on n vertices without a subgraph H? How does a graph look like if it has this extremal edge number? The forbidden graph in this article is a clique-path: a path of length k where each edge is extended to an r-clique, r >2. We determine both the extremal number and the extremal graphs for sufficiently large n.
Spectral extremal problems for degenerate graphs
本文研究退化图族的谱极值问题，利用图的覆盖与独立覆盖建立谱稳定性结果，证明了一般性定理以刻画谱极值图，并揭示了极值图与谱极值图之间的相关性。
The \emph{blow-up} of a graph $H$ is the graph obtained from replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blow-ups of paths. We determined the extremal number and found the extremal graphs for the blow-ups of cycles and a large class of trees, when $n$ is sufficiently large. This generalizes their results. The additional aim of our note is to draw attention to a powerful tool…
Extremal regular graphs of given chromatic number
该研究定义了阶数最小的给定色数$r$-正则图即极$(r|χ)$-图，证明三类图是极图，还研究了极Cayley $(r|χ)$-图并给出相关构造。
This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.
A general expression for calculating the bond incident degree (BID) indices of certain triangular chain graphs is derived. The extremal triangular chain graphs with respect to several well known BID indices are also characterized over a particular collection of triangular chain graphs.
On edge-ordered graphs with linear extremal functions
该研究刻画了具有线性极值函数的连通边有序图，证明其余连通边有序图极值函数为Ω(n log n)，还拓展了边有序路径极值函数的研究范围。
The Turán type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Turán type extremal problem asks to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without containing a linear forest as a subgraph and characterize all corresponding extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without containing $k\cdot P_3$ as a subgraph are obtained and all extremal graphs a…
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest matchings of size $k$; indeed either the lex graph or the colex graph is such an extremal example. In this paper we consider the problem of maximizing the number of matchings in the class of threshold graphs. We prove that the minimizers are what we call \emph{almost alternating threshold graphs}. We also discuss a problem with a similar flavor: which threshold gr…
Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corresponding extremal graphs are also determined.
An extremal graph for a given graph $H$ is a graph with maximum number of edges on fixed number of vertices without containing a copy of $H$. The $k$-th power of a path is a graph obtained from a path and joining all pair of vertices of the path with distance less than $k$. Applying a deep theorem of Simonovits, we characterize the extremal graphs of the $k$-th power of paths. This settles a conjecture posed by Xiao, Katona, Xiao and Zamora in a stronger form.
We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We completely solve this problem when $H$ is a clique, as well as more generally when $H$ is any regular graph with degree at least about half its number of vertices. We also characterize the extremal graphs when $H$ is an Erdős-Rényi random graph. The extremal structures turn out to have the similar form as the conjectured extremal structures for a well-studied b…
We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.
We investigate extremal graphs related to the game of Cops and Robbers. We focus on graphs where a single cop can catch the robber; such graphs are called cop-win. The capture time of a cop-win graph is the minimum number of moves the cop needs to capture the robber. We consider graphs that are extremal with respect to capture time, i.e. their capture time is as large as possible given their order. We give a new characterization of the set of extremal graphs. For our alternative approach we assign a rank to each vertex of a graph, and then study which configurations of ranks are possible. We p…
This paper is devoted to present two counterexamples to the theorem from \cite{MK} Maria R., Katherine T. M., Bernardo S. M., Extremal graphs with bounded vertex bipartiteness number, Linear Algebra Appl. 493 (2016) 28-36. Moreover, the corrected theorem and proof are presented.
Given two graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a generalized Turán problem, i.e., an $F$-free $n$ vertex graph $G$ that for some $H$ contains exactly $\mathrm{ex}(n,H,F)$ copies of $H$.
Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n, \mathcal{H})$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containing no graph from the family $\mathcal{H}$ as a subgraph. We show that for every rational number $r$ between $1$ and $2$, there is a family of graphs $\mathcal{H}_r$ such that $\textrm{ex}(n, \mathcal{H}_r) = Θ(n^r)$. This solves a longstanding problem in the area of extremal graph theory.
In this expository paper, we present a motivated construction of large graphs not containing a given complete bipartite subgraph. The key insight is that the algebraic constructions yield very non-smooth probability distributions.
Spectral Extremal Graphs of Planar Graphs with Fixed Size
本文研究固定边数外平面图与平面图的谱极值问题，证明外平面图极值图为星图，并给出平面图在边数奇偶不同时的谱极值图结构。
The transmission ${\rm Tr}_G(v)$ of a vertex $v$ of a connected graph $G$ is the sum of distances between $v$ and all other vertices in $G$. $G$ is a stepwise transmission irregular (STI) graph if $|{\rm Tr}_G(u) - {\rm Tr}_G(v)| =1$ holds for each edge $uv \in E(G)$. In this paper, extremal results on STI graphs with respect to the size and different metric properties are proved. Two extremal families appear in all the cases, balanced complete bipartite graphs of odd order and the so called odd hatted cycles.
The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal values given natural constraints. In this paper, we prove that the graphs with given size that minimize the first degree-based entropy are the colex graphs.
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.
On extremal (almost) edge-girth-regular graphs
该研究聚焦边围长正则图（egr-图）的极值问题，证明有限平面关联图衍生的部分图为三元组$(k,g,λ)$的极值egr-图，并给出egr-图阶数的新下界。
The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified classes of graphs. We develop tools for treating such problems and obtain some partial results. Using calculus, we show that an extremal graph ``should'' have a small number of distinct eigenvalues. However, we also present data that shows in many cases that extremal graphs can have a large number of distinct eigenvalues.
This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also pr…
The graph entropy describes the structural information of graph. Motivated by the definition of graph entropy in general graphs, the graph entropy of hypergraphs based on Laplacian degree are defined. Some results on graph entropy of simple graphs are extended to k-uniform hypergraphs. Using an edge-moving operation, the maximum and minimum graph entropy based on Laplacian degrees are determined in k-uniform hypertrees, unicyclic k-uniform hypergraphs, bicyclic k-uniform hypergraphs and k-uniform chemical hypertrees, respectively, and the corresponding extremal graphs are determined.
We prove an upper bound for the number of edges a C4-free graph on q^2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q.
Extremal problems involving independent sets are much studied. Two of the most important extremal problems in this context are concerned with the sharp upper bounds for the number of independent sets of fixed size and the independence number. In literature, these sharp upper bounds are derived in completely different contexts. In this paper, we show that both of these sharp upper bounds can be derived by considering Lex graphs. More exactly, they depend on two parameters of a sequence defined on them.
Extremal eigenvalues of outerplanar graphs
本文研究外平面图的极值特征值，通过刻画极大二部外平面图的结构，完全确定了阶数至少55的二部外平面图的最大谱半径和一般外平面图的最小最小特征值。
In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by Bollobás, it is very difficult for general $k$. Recently, Tait and Tobin [J. Combin. Theory Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of $n$-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without $k$ edge-disjoint cycles. We first determine the maximum number of edges in a plan…
In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index τ(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
Turán-Type Extremal Results for Distance-$k$ Graphs
研究距离图的Turán型极值问题，确定无三角形的n顶点图中距离三的顶点对最大数，解决Tyomkyn和Uzzell的猜想。同时确定距离二的顶点对最大数并完全刻画极值图。
Extremal graphs for odd-ballooning of paths and stars
本文利用Simonovits逐步归纳引理，统一确定了q≥3时星与路径的奇膨胀图的极值图，完善了此前分情况的相关研究结果。
On extremal graphs with respect to the ABS index
本文研究了关于绝对谱指数的极值图问题，解决了具有特定顶点数、割顶点数和顶点连通度的图类的最大绝对谱指数问题。
The Turán type numbers for graphs without 3-cycles and 4-cycles are determined for vertex numbers from 40 to 49 inclusive. Hence, now, 43 of the first 50 numbers of OEIS A006856 are known. Estimates for the remaining seven numbers are presented.
Define f(n,p) to be the maximum number of edges in a graph on n vertices with p perfect matchings. Dudek and Schmitt proved there exist constants n_p and c_p so that for even n >= n_p, f(n,p) = (n^2)/4+c_p. A graph is p-extremal if it has p perfect matchings and (n^2)/4+c_p edges. Based on Lovasz's Two Ear Theorem and structural results of Hartke, Stolee, West, and Yancey, we develop a computational method for determining c_p and generating the finite set of graphs which describe the infinite family of p-extremal graphs. This method extends the knowledge of the size and structure of p-extremal…
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the quantity $i(G)^{1/v(G)}$, the number of independent sets in $G$ normalized exponentially by the size of $G$? What if $i(G)$ is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.
Let $ρ(G)$ denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 <= g <=n. It is proved that $ρ(G) \leq \frac{n}{g}(m-n+1)$ and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.
Uniformity of extremal graph-codes
研究极值图码及其与密度多项式Hales-Jewett猜想相关问题，揭示极值离散结构的强伪随机行为。
In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conje…
We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and Bollobas, but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.
The famous lower bound $α(G)\geq \sum_{u\in V(G)}\frac{1}{d_G(u)+1}$ on the independence number $α(G)$ of a graph $G$ due to Caro and Wei is known to be tight if and only if the components of $G$ are cliques, and has been generalized several times in the context of large degenerate subsets and small dynamic monopolies. We characterize the extremal graphs for a generalization due to Ackerman, Ben-Zwi, and Wolfovitz. Furthermore, we give a simple proof of a related bound concerning partial incentives due to Cordasco, Gargano, Rescigno, and Vaccaro, and also characterize the corresponding extrema…
Extremal Peisert-type graphs without the strict-EKR property
本研究针对不具有严格EKR性质的边数最少的极值Peisert型图，确定了各素数幂q对应的边数，证明平方或立方q下极值图的唯一性并分类最大团，还验证了平方q下的相关界的紧性。
Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs
本文利用正则方法和Füredi定理，证明了在特定条件下，无符号拉普拉斯谱极值图的集合是Turán极值图集合的子集。
Extremal distance spectra of graphs and essential connectivity
本文研究图的本质连通度与距离谱半径的极值关系，确定了给定本质连通度下具有最小距离谱半径的图、连通图及强连通有向图的极值结构。
A fundamental problem in pattern avoidance is describing the asymptotic behavior of the extremal function and its generalizations. We prove an equivalence between the asymptotics of the graph extremal function for a class of bipartite graphs and the asymptotics of the matrix extremal function. We use the equivalence to prove several new bounds on the extremal functions of graphs. We develop a new method to bound the extremal function of hypergraphs in terms of the extremal function of their associated multidimensional matrices, improving the bound of the extremal function of $d$-permutation hy…
According to Paul Erdős [Some notes on Turán's mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Turán who "created the area of extremal problems in graph theory". However, without a doubt, Paul Erdős popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and all over the world. In fact, most of the early contributions in this field can be traced back to Paul Erdős, Paul Turán, as well as their collaborators and students. Paul Erdős also established the probabi…
Let $\G$ be an oriented graph of order $n$ and $\a_1,\a_2,..., \a_n$ denote all the eigenvalues of the skew-adjacency matrix of $\G.$ The skew energy $\displaystyle{\cal E}_s(\G)= \sum_{i=1}^{n} |\a_i|.$ In this paper, the oriented unicyclic graphs with minimal and maximal skew energy are determined.
We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young tableaux, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory, but they are easy to find for specific …
In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$ denote the family of $d$--regular graphs. The general problem studied in this paper is to determine $$\inf_{G\in \mathcal{G}_d}\hom(G,A)^{1/v(G)}.$$ It turns out that in many instances the infimum is not achieved by a finite graph, but a sequence of graphs with girth (i. e., length of the shortest cycle) tending to infinity. In other words, the optimization pr…
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers. We then determine the asymptotic order of the largest size of a graph with given general or monophonic position number, classifyin…
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every $k$, we also determine the connected graphs that are extremal with respect to the number of $k$-vertex connected induced subgraphs. We show that, in contrast to the minimum which is…
In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We show the connection between minimizing the total distance and maximizing the size under the same conditions. We also prove some asymptotically optimal bounds for the maximum total distance.
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