In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $λ_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for all $k>2,$ and even tighter bounds are obtained for the $k$'th largest singular value $λ_{k}^{\ast}(G).$
Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large can $λ_{k}(G)+λ_{k}(\bar{G})$ be$?$
These constructions are successful also in another open question: How large can the Ky Fan norm $λ_{1}^{\ast}(G)+...+λ_{k}^{\ast }(G)$ be $?$ Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs.
In the final section, several results and problems are restated for $(-1,1)$-matrices, which seem to provide a more natural ground for such research than graphs.
Many of the results in the paper are paired with open questions and problems for further study.
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.
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.
Let $G$ be a connected simple graph on $n$ vertices. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$ and $ρ_{n-1}(G)$ be the second least eigenvalue of $\mathcal{L}(G)$. Denote by $ν(G)$ the independence number of $G$. Recently, the paper [Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, arXiv:1912.13227] discussed the graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$. However, there is one remaining case (graphs with $ρ_{n-1}(G)\neq 1$ and $ν(G)= 2$) not considered. In this paper, we focus on cographs and graphs wi…
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 eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and cha…
It is a well-known fact that a graph of diameter $d$ has at least $d+1$ eigenvalues. Let us call a graph \emph{$d$-extremal} if it has diameter $d$ and exactly $d+1$ eigenvalues. Such graphs have been intensively studied by various authors. %Much attention has been devoted to the study of graphs that are extremal with respect to this relation: \emph{i.e} have diameter $d$ and exactly $d+1$ distinct eigenvalues. A graph is \emph{split} if its vertex set can be partitioned into a clique and a stable set. Such a graph has diameter at most $3$. We obtain a complete classification of the connected …
A complex unit gain graph ($\mathbb{T}$-gain graph), $Φ= (G, φ)$ is a graph where the function $φ$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. A $ \mathbb{T} $-gain graph $Φ$ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is $1$. Signed graphs are special cases of $\mathbb{T}$-gain graphs. The adjacency matrix of $Φ$, denoted by $ \mathbf{A}(Φ)$ is defined canonically. The gain Laplacian for $Φ$ is defined as $\mathbf{L}(Φ) = \mathbf{D}(Φ) - \mathbf{A}(Φ)$, where $\mathbf{D}…
Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$, $i_{0}(G_w)$, respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order $n$ with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order $n$ with at most two positive, two negative and at least $n-4$ zero eigenvalues are identified, respectively.
In this paper, we study extremal problems of Steklov eigenvalues on combinatorial graphs by extending Friedman's theory [Duke Math. J. 69 (1993), no. 3, 487--525] of nodal domains for Laplacian eigenfunctions to Steklov eigenfunctions, and solve an extremal problem for Steklov eigenvalues on combinatorial graphs that is an analogue of the extremal problem solved by Friedman [Duke Math. J. 83 (1996), no. 1, 1--18.] for Laplacian eigenvalues. More precisely, we mainly show that the minimum of the $i^{\rm th}$ Steklov eigenvalue on a connected combinatorial graph with $n$ vertices is essentially …
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.
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.
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…
Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdière parameter, excluded minors, and the spectral radius, J. Combin. Theory Ser. A 166 (2019) 42--58] determined the maximum spectral radius and characterized the unique extremal graph for $K_r$-minor free graphs of sufficiently large order $n$, he also made great progress on $K_{s,t}$-minor free graphs and posed a conjecture: Let $2\leq s\leq t$ and $n-s+1=pt+q$, where $n$ is sufficiently large and $1\leq q\leq t.$ Then $K_{s-1}\nabla (pK_t\cup K_q)$ is the unique extremal graph w…
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.
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.
Let G be a simple graph on $n$ vertices and $e(G)$ edges. Consider $Q(G) = D + A$ as the signless Laplacian of $G$, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of the vertices degree of $G$. Let $q_1(G)$ and $q_2(G)$ be the first and the second largest eigenvalues of $Q(G),$ respectively, and denote by $S_{n}^{+}$ the star graph plus one edge. In this paper, we prove that inequality $q_1(G)+ q_2(G) <= e(G)+3$ is tighter for the graph $S_{n}^{+}$ among all firefly graphs and also tighter to $S_{n}^{+}$ than to the graphs $K_{k} \vee \overline{K_{n-k}}$ recently presented by…
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 $H$.
School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）；School of Mathematics, East China University of Science and Technology（华东理工大学数学系）；School of Mathematics and Finance, Chuzhou University（滁州学院数学与金融学院）
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.
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.
Let $G$ be a simple graph and $Q(G)$ be the signless Laplacian matrix of $G$. Let $S_α(G)$ be the sum of the $α$-th powers of the nonzero eigenvalues of $Q(G)$. We disprove two conjectures by You and Yang on the extremal values of $S_α(G)$ among bipartite graphs and among graphs with bounded connectivity.
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…
Turán-Type Extremal Results for Distance-$k$ Graphs
Beijing Jiaotong University（北京交通大学）；Jagiellonian University（雅盖隆大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育国家研究所阿尔弗雷德·雷尼数学研究所）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Extremal Graph Theory aims to determine bounds for graph invariants as well as the graphs attaining those bounds. We are currently developping PHOEG, an ecosystem of tools designed to help researchers in Extremal Graph Theory. It uses a big relational database of undirected graphs and works with the convex hull of the graphs as points in the invariants space in order to exactly obtain the extremal graphs and optimal bounds on the invariants for some fixed parameters. The results obtained on the restricted finite class of graphs can later be used to infer conjectures. This database also allows …
The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.
We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that `every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints.
In this paper we are concerned with various graph invariants (girth, diameter, expansion constants, eigenvalues of the Laplacian, tree number) and their analogs for weighted graphs -- weighing the graph changes a combinatorial problem to one in analysis. We study both weighted and unweighted graphs which are extremal for these invariants. In the unweighted case we concentrate on finding extrema among all (usually) regular graphs with the same number of vertices; we also study the relationships between such graphs.
