Vladimir Nikiforov
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.
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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。
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 …
Extremal graphs for the $k$-th eigenvalue
本文研究Hong问题中λₖ(G)的最优上界，刻画Sivashankar定理的等号情形，还给出λ₃和λ₄极值图的显式组合描述。
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 $\mathbf{D}…
Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues
本文研究了前两个最大无符号拉普拉斯特征值之和的极图问题，证明了在具有$e(G)$条边的图中，$K^+_{1,e(G)-1}$是使$f(G)$取得最小值的唯一图。
Trees with extremal Laplacian eigenvalue multiplicity
本文完整刻画了拉普拉斯特征值重数达到悬挂顶点数减一的树，证明其要么为路径，要么所有悬挂顶点间距离满足特定同余条件，并由此给出特征值1达到该极值重数的充要条件。
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.
Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The Sombor index of the graph $G$ is a degree-based topological index, defined as $$SO(G)=\sum_{uv \in E}\sqrt{d(u)^2+d(v)^2},$$ in which $d(x)$ is the degree of the vertex $x \in V$ for $x=u, v$. In this paper, we characterize the extremal trees with a given degree sequence that maximizes the Sombor index.
On the nontrivial extremal eigenvalues of graphs
本文通过特征函数分析，对有限非二部Cayley图给出了非平凡特征值远离±1的精细定量刻画，并建立了最小归一化邻接特征值与-1之间间隙的新显式下界，改进了基于边扩展的已知结果。
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.
Extremal Bounds on the Sigma and Albertson Indices for Non-Decreasing Degree Sequences
本文研究了非递增度序列树的阿尔伯特森和西格马不规则指数的极值界限，通过毛毛虫树等关键配置，提出了基于最大度数、平均度数和辅助序列参数的新界限，并验证了这些界限的紧密性。
The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.
Let $r(u,v)$ be the resistance distance between two vertices $u, v$ of a simple graph $G$, which is the effective resistance between the vertices in the corresponding electrical network constructed from $G$ by replacing each edge of $G$ with a unit resistor. The degree resistance distance of a simple graph $G$ is defined as ${D_R}(G) = \sum\limits_{\{u,v\} \subseteq V(G)} {[d(u) + d(v)]r(u,v)},$ where $d(u)$ is the degree of the vertex $u$. In this paper, the bicyclic graphs with extremal degree resistance distance are strong-minded. We first determine the $n$-vertex bicyclic graphs having pre…
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.
Extremal degree-based indices of general polyomino chains via dynamic programming
本文通过动态规划方法研究一般多形链的极值度数基于指数，解决了一个关于广义Randić指数的开放问题。
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.
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 …
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 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.
A note on extremal Sombor indices of trees with a given degree sequence
该研究将给定度序列的树的Sombor指数极值问题纳入已有成果框架，证明贪心树的Sombor指数最小，交替贪心树的该指数最大。
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.
The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate achemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. The tree that minimizes the Wiener index among trees of given maximal degree was studied. We characterize trees that achieve the maximum and minimum Wiener index, given the number of vertices and the degree sequence.
The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solva…
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…
Extremal Polynomial Norms of Graphs
本文定义了一种基于完全齐次对称多项式的图范数，证明其在连通图中由路径和完全图分别达到最小和最大，在树中由路径和星图分别达到最小和最大。
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 …
For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $π=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invarian…
Extremal $ABS$ Spectral Radius in Bicyclic and Bipartite Unicyclic Graphs
本文研究了双环图和二分单环图中ABS谱半径的极值问题，确定了最大和次大谱半径的图结构。
Convex combination of first and second eigenvalues of trees
本文研究了树的首和第二特征值的凸组合问题，确定了在不同参数下达到极值的树结构。
We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants (girth, diameter), and spectral invariants (the determinant of the Laplace operator, or complexity; bottom non-zero eigenvalue of the Laplace operator). We show that almost all of these functions are, surprisingly, convex, and we characterize the valuations extremizing these invariant
Extremal eigenvalues with respect to graph minors
本文研究了图矿的极值特征值问题，通过统一现象证明了图中存在矿或覆盖书结构，并利用稳定性方法、谱技术及结构分析扩展了平面图、外平面图和K_{s,t}矿自由图的猜想。
In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $π=(d_1, ..., d_5, 1,..., 1)$ have been characterized.
This paper considers the degree-diameter problem for extremal and largest known undirected circulant graphs of degree 2 to 9 of arbitrary diameter. As these graphs are vertex transitive it is possible to define their distance partition. The number of vertices in each level of the distance partition is shown to be related to an established upper bound for the order of Abelian Cayley graphs. Furthermore these graphs are all found to have odd girth which is maximal for their diameter. Therefore the type of each vertex in a level may be well-defined by the number of adjacent vertices in the preced…
On vertex-transitive distance-regular covers of complete graphs with an extremal smallest eigenvalue
本文研究完全图的阿贝尔距离正则覆盖，利用秩至多3的置换群分类，刻画了具有极值最小特征值且自同构群顶点传递、弧集上至多两个轨道的覆盖族。
Closed-Form Analysis and Extremal Bounds of Albertson and Sigma Indices in Trees with Prescribed Degree Sequences
本文研究了具有给定度序列的树的阿尔伯特森指数和西格玛指数的闭式表达式及极值界限，为不规则性度量提供了新的分析框架。
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.
We study the eigenvectors of Laplacian matrices of trees. The Laplacian matrix is reduced to a tridiagonal matrix using the Schur complement. This preserves the eigenvectors and allows us to provide fomulas for the ratio of eigenvector entries. We also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and Perron values. The results are then applied to the Fiedler vector. Here we locate the extremal entries of the Fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path.
In this paper extremal problems for uniform hypergraphs are studied in the general setting of hereditary properties. It turns out that extremal problems about edges are particular cases of a general analyic problem about a recently introduced graph parameter. The paper builds a basis for the systematic study of this parameter and illustrates a range of various proof tools. It is shown that extremal problems about the number of edges of uniform hypergraphs are asymptotically equivalent to extremal problems about the largest eigenvalue; this result is new even for 2-graphs. Several concrete prob…
Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $φ$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{φ(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randić index, the Zagreb index, the…
Extremal chemical graphs of maximum degree at most 3 for 33 degree-based topological indices
本文研究最大度不超过3的化学图，针对33种基于度的拓扑指标刻画其极值图，发现仅需五个图族即可覆盖其中29个指标的极值情形，表明这些指标的极值性质差异很小。
For a degree sequence $d:d_1\geq \cdots \geq d_n$, we consider the smallest chromatic number $χ_{\min}(d)$ and the largest chromatic number $χ_{\max}(d)$ among all graphs with degree sequence $d$. We show that if $d_n\geq 1$, then $χ_{\min}(d)\leq \max\left\{ 3,d_1-\frac{n+1}{4d_1}+4\right\}$, and, if $\sqrt{n+\frac{1}{4}}-\frac{1}{2}>d_1\geq d_n\geq 1$, then $χ_{\max}(d)=\max\limits_{i\in [n]}\min\left\{ i,d_i+1\right\}$. For a given degree sequence $d$ with bounded entries, we show that $χ_{\min}(d)$, $χ_{\max}(d)$, and also the smallest independence number $α_{\min}(d)$ among all graphs wit…
The extremal generalised Randić index for a given degree range
本文通过更一般命题给出O-Shi关于Randić指数下界结论的简短证明，并据此分类任意度范围内使广义Randić指数取极值的图，精确刻画行为转变。
We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.
We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\b…
Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $A(G)$ and $D(G)$ denote, respectively, the adjacency and the diagonal matrix of the vertex degrees of $G$. Let $q_1\geq q_2\geq \dots\geq q_n\geq 0$ be the signless Laplacian eigenvalues of $G$. The largest signless Laplacian eigenvalue $q_1$ is called the signless Laplacian spectral radius or $Q$-i…
Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $Δ$. Let $λ_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the largest eigenvalue of non-regular graphs, J. Combin. Theory Ser. B, 97 (2007) 1010--1018] asserts that \[ \lim_{n\to\infty} \frac{n^2(Δ-λ_1(G))}{Δ-1} = π^2 \] for each fixed $Δ$. Concerning an important structural property of the extremal graphs $G$, Liu and Li present another conjecture which states that $G$ has degree sequence $Δ,\ldots,Δ,δ$. Here, $δ=Δ-1$ or $δ=Δ-2$…
Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size
本文研究了细分边对图$f$-谱半径的影响，推导了给定阶数和边数极值图的必要条件，并应用广义Lu-Man方法求出了$n\geq8$时具有最小$f$-谱半径的双圈图。
Extremal Degree Irregularity Bounds for Albertson and Sigma Indices in Trees and Bipartite Graphs
本文研究了树和双图中Albertson和Sigma指数的极值界限，推导了不规则性度量的显式公式和上界，通过度数序列的不等式推进了对双图极值不规则性行为的理解。
ChemicHull: an online tool for determining extremal chemical graphs of maximum degree at most 3 for any degree-based topological indices
ChemicHull是一种在线工具，用于确定任意度数基于拓扑指数的极值化学图，通过线性优化在多面体框架内分析化学图的极值特性。
Degree Sequence Optimization and Extremal Degree Enumerators
本文通过非线性变换将度序列优化转化为度枚举子多面体问题，刻画了完全图及部分完全二部图的极值顶点，并据此提出更简单快速的优化算法。
On the degree-$2$ Siegel theta series of extremal even unimodular lattices of ranks $48$, $72$, $96$, and $120$
利用Igusa结构定理和深度过滤，研究极值偶幺模格子的2次Siegel theta级数，证明在秩48、72和96时由极值性唯一确定，在秩120时相差$\chi_{10}^6$的整数倍。
Extremal chromatic bounds for distance Laplacian eigenvalues
研究距离拉普拉斯矩阵的极值色数界限，提出色类主导原理，改进色数阈值下的特征值分布定理，并通过邻域压缩获得显式色数条件。
Let F(G) be a fixed linear combination of the k extremal eigenvalues of a graph G and of its complement. The problem of finding max{F(G):v(G)=n} generalizes a number of problems raised previously in the literature. We show that the limit max{F(G):v(G)=n}/n exists when n tends to infinity. We also answer a question of Gernert about the sum of the two maximal eigenvalues of a graph.
Extremal spectral behavior of weighted random $d$-regular graphs
本文研究带随机边权重的随机 $d$-正则图，刻画极值谱随边权尾部的相变，并证明极值特征向量局部化。
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.
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