Sebastian M. Cioabă
Let $G$ be an irregular graph on $n$ vertices with maximum degree $Δ$ and diameter $D$. We show that Δ-λ_1>\frac{1}{nD} where $λ_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.
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Let $λ^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $Δ$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(Δ-λ^{*}))/(Δ-1)=π^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $λ_{1}(G)$, we present a …
A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{π^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum…
Maximum ratio of (graph) irregularities
本文研究了图和树中σ_t不规则度与σ不规则度的最大比值，证明了其界限并回答了Filipovski等人提出的猜想。
For a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s$-th (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.
On the transmission irregular trees with the maximum Wiener index
本文研究了在给定阶数的传输不规则树中最大化维纳指数的问题，针对奇数和几乎所有偶数阶数给出了唯一解。
On the maximum $σ$-irregularity of trees with given order and maximum degree
本文研究了具有给定阶数和最大度数的树的最大σ不规则性，通过线性规划方法得到精确上界并明确极值树的结构。
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。
Trees with maximum $σ$-irregularity under a prescribed maximum degree 6
研究在最大度数6限制下，确定最大σ不规则性树的结构和度数分布。
Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $Δ$, and $d=\left\lfloor \frac{Δm}{Δn-m}\right\rfloor$, we show $$irr(G)\leq d(d+1)n+\frac{1}Δ\left(Δ^2-(2d+1)Δ-d^2-d\right)m.$$
2-Distance Coloring of 4-Irregular Planar Graphs
本文研究4-非正则平面图的2-距离着色问题，将其2-距离色数上界从Zhu证明的13改进为10，推进了平面图着色相关理论的研究。
The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i(e)=|d(u)-d(v)|$, where $d(\cdot)$ is the vertex degree. The irregularity $I(G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was introduced by Albertson who proved that $I(G) \leq \frac{n^{3}}{27}$ (where $n=|V(G)|$) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2011. Our bound involves the Laplacian spectral …
The distance spectral radius of $k$-uniform hypertrees with given number of vertices of maximum degree
本文通过嫁接变换研究给定大小、最大度及最大度顶点数的$k$-一致超树，确定了具有最大距离谱半径的超图结构。
Let $m:=|E(G)|$ sufficiently large and $s:=(m-1)/3$. We show that unless the maximum degree $Δ> 2s$, there is a weighting $w:E\cup V\to \{0,1,...,s\}$ so that $w(uv)+w(u)+w(v)\ne w(u'v')+w(u')+w(v')$ whenever $uv\ne u'v'$ (such a weighting is called {\em total edge irregular}). This validates a conjecture by Ivanco and Jendrol' for large graphs, extending a result by Brandt, Miskuf and Rautenbach.
Let $H$ be an $r$-uniform hypergraph on $n$ vertices and $m$ edges, and let $d_i$ be the degree of $i\in V(H)$. Denote by $\varepsilon(H)$ the difference of the spectral radius of $H$ and the average degree of $H$. Also, denote \[ s(H)=\sum_{i\in V(H)}\left|d_i-\frac{rm}{n}\right|,~ v(H)=\frac{1}{n}\sum_{i\in V(H)}d_i^{\frac{r}{r-1}}-\left(\frac{rm}{n}\right)^{\frac{r}{r-1}}. \] In this paper, we investigate the irregularity of $r$-uniform hypergraph $H$ with respect to $\varepsilon(H)$, $s(H)$ and $v(H)$, which extend relevant results to uniform hypergraphs.
On the maximum spectral radius of planar graphs
本文研究了具有固定顶点数的平面图的最大谱半径，给出了基于阶数的紧致界，并证明了在包含支配顶点且阶数 $n \geq 48$ 的平面图中，$P_{2}$ 与 $P_{n-2}$ 的联图取得最大谱半径。
Consider a simple graph $G$. We call a labeling $w:E(G)\cup V(G)\rightarrow \{1, 2, \dots, s\}$ (\textit{total vertex}) \textit{product-irregular}, if all product degrees $pd_G(v)$ induced by this labeling are distinct, where $pd_G(v)=w(v)\times\prod_{e\ni v}w(e)$. The strength of $w$ is $s$, the maximum number used to label the members of $E(G)\cup V(G)$. The minimum value of $s$ that allows some irregular labeling is called \textit{the total vertex product irregularity strength} and denoted $tvps(G)$. We provide some general bounds, as well as exact values for chosen families of graphs. Keyw…
Let G be a graph of n vertices and m edges, and let G has no cycles of length 4. We give upper bounds on the adjacency spectral radius of G in terms of n and m.
Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $κ(G) \leq k$ and showed that among those graphs, the maximum spectral radius is obtained uniquely at $K_k^n$, which is the graph obtained by joining $k$ edges from $k$ vertices of $K_{n-1}$ to an isolated vertex. In this paper, we study the spectral radius of graphs of order $n$ with $κ(G)\leq k$ and minimum degree $δ(G)\geq k $. We show that among those graphs, the maximum spectral radius is obtained uniquely at $K_{…
We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of Stevanovic, Zhang, and Alon and Sudakov.
Extremizing antiregular graphs by modifying total $σ$-irregularity
针对总$σ$-不规则性最大值由少度分裂图取得的反直觉结论，修正不规则性不变量形式，研究其在反正则图上取最大值的条件，覆盖多类图并提出相关问题与猜想。
Connected irregular cospectral graphs with distinct Lovász numbers
针对每个n≥10构造连通非正则非同构同谱图对，它们共享四种谱和四个组合不变量但Lovász数不同，最小阶数为10。
If $A$ is an independent set of a graph $G$ such that the vertices in $A$ have different degrees, then we call $A$ an irregular independent set of $G$. If $D$ is a dominating set of $G$ such that the vertices that are not in $D$ have different numbers of neighbours in $D$, then we call $D$ an irregular dominating set of $G$. The size of a largest irregular independent set of $G$ and the size of a smallest irregular dominating set of $G$ are denoted by $α_{ir}(G)$ and $γ_{ir}(G)$, respectively. We initiate the investigation of these two graph parameters. For each of them, we obtain sharp bounds…
Spectral radius and maximum matching covered graphs with bounded matching number
本文给出了具有有界匹配数的图成为最大匹配覆盖图的尖锐谱半径条件。
A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of a graph of order $n$ that does not contain subgraphs of particular kind. In this paper, we study the Brualdi-Solheid-Turán type problem for wheel-free graphs, and we determine the maximum (signless Laplacian) spectral radius of a wheel-free graph of order $n$. Furthermore, we characterize the extremal graphs.
Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $λ_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for $D_{\max}(G)-λ_1(G)$, and characterize the extremal graphs attaining the bound. As a corollary, we solve a conjecture posed by Liu, Shu and Xue.
For $0\le α\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_α(G)=αD(G)+(1-α)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $α$-spectral radius of $G$ is the largest eigenvalue of $A_α(G)$. We give upper bounds for $α$-spectral radius for unicyclic graphs $G$ with maximum degree $Δ\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $α$-spectral radius…
Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear Algebra Appl. 436(2012), 4512--1829] showed that the spectral radius of $G^σ$ is the same for every orientation $σ$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. This paper, by using the Kelmans transformation, …
We study the effect of three types of graft transformations to increase or decrease the distance spectral radius of uniform hypergraphs, and we determined the unique $k$-uniform hypertrees with maximum, second maximum, minimum and second minimum distance spectral radius, respectively.
Research on the relationship of the (signless Laplacian) spectral radius of a graph with its structure properties is an important research project in spectral graph theory. Denote by $ρ(G)$ and $q(G)$ the spectral radius and the signless Laplacian spectral radius of a graph $G$, respectively. Let $k\ge 0$ be a fixed integer and $G$ be a graph of size $m$ which is large enough. We show that if $ρ(G)\ge\sqrt{m-k}$, then $C_4\subseteq G$ or $K_{1,m-k}\subseteq G$. Furthermore, we prove that if $q(G)\ge m-k$, then $K_{1,m-k}\subseteq G$. Both these two results extend some known results.
In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for some } h\in E(\mathcal{H})\}$. A graph is \textit{outerplanar} if it can be embedded in the plane such that all its vertices lie on the outer face. A $3$-uniform hypergraph $\mathcal{H}$ is called \textit{outerplanar} if its shadow has an outerplanar embedding such that every hyperedge of $\mathcal{H}$ is the vertex set of an interior triangular face of the sha…
On the maximum spectral radius of connected graphs with a prescribed order and size
本文研究了具有指定阶数和大小的连通图的最大谱半径问题，给出了在特定条件下谱半径最大化的结果。
Let $G$ be a simple graph with vertex set $V(G) = \{v_1 ,v_2 ,\cdots ,v_n\}$. The Harary matrix $RD(G)$ of $G$, which is initially called the reciprocal distance matrix, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $\frac{1}{d_{ij}}$ if $i\not=j$ and $0$ otherwise, where $d_{ij}$ is the distance of $v_i$ and $v_j$ in $G$. In this paper, we characterize graphs with maximum spectral radius of Harary matrix in three classes of simple connected graphs with $n$ vertices: graphs with fixed matching number, bipartite graphs with fixed matching number, and graphs with given number of cut …
For $0\leq α< 1$, the $\mathcal{A}_α$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_α(G):=α\mathcal{D}(G)+(1-α) \mathcal{A}(G)$, where $\mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $\mathcal{A}_α$-spectral radius and an average degree $\frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the …
Unbalanced signed bipartite graphs containing no negative $C_4$ with maximum spectral radius
本文研究了不含负四边形的不平衡带符号二分图的最大谱半径问题，确定了在固定二分图大小和阶数下唯一具有最大谱半径的图。
Some results on $σ_{t}$-irregularity
本文研究图不变量sigma总指标，发现其等于n²倍度方差，据此刻画了连通图和完全二分图中具有最大sigma总指标的图，并给出了上下界及与图能量的关系，最后提出两个猜想。
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal matrix of the degrees of $G$. In 2017, Nikiforov defined the $A_α$-matrix of $G$ as \begin{equation*} A_α(G)=αG)+(1-α)A(G), \end{equation*}d where $α\in[0,1]$ is an arbitrary real number. The largest eigenvalue of $A_α(G)$ is called the $A_α$-spectral radius of $G$. Let $n$, $t$, $k$ be positive integers, satisfying $t\geq1$, $k\geq2$, $n\geq k+2$, and $n\equiv k$ (mod $2$). In this paper, for $α\in[0,\frac{1}{2}]$, we determine the extremal graphs with the maximum $A_α$-spectral radius among all $t$-connected graphs …
Decomposability of regular graphs to $4$ locally irregular subgraphs
针对局部不规则边分解猜想的反例，本文证明所有度数 $d\geq 54000$ 的 $d$-正则图均可分解为 $4$ 个局部不规则子图。
A \emph{locally irregular graph} is a graph whose adjacent vertices have distinct degrees. We say that a graph $G$ can be decomposed into $k$ locally irregular subgraphs if its edge set may be partitioned into $k$ subsets each of which induces a locally irregular subgraph in $G$. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree $3$, every connected graph can be decomposed into $3$ locally irregular subgraphs. Using a combination of a probabilistic approach and some kn…
List Coloring of the Square of 4-Irregular Graphs
研究4-不规则图平方的列表染色问题，通过证明得出对于此类图\(\chi_l(G^2) \leq 11\)，并给出例子说明此界是紧的，对克兰斯顿和金的相关研究进行了补充。
The maximum spectral radius of $θ_{2,2,3}$-free graphs with given size
本文研究了给定边数$m$的$\theta_{2,2,3}$-自由图的最大谱半径，并刻画了达到最大谱半径的极值图。
Consider a graph $G=(V,E)$ without isolated edges and with maximum degree $Δ$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$. For any integer $r\geq 2$, the least $k$ admitting the existence of such $c$ attributing distinct weighted degrees to any two different vertices at distance at most $r$ in $G$ is called the $r$-distant irregularity strength of $G$ and denoted by $s_r(G)$. This graph invariant provides a natural link between the well known 1--2--3 Conjecture and irregularity strength of g…
On the maximum $A_α$-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices
本文研究了单圈图和双圈图的 $A_α$-谱半径极值问题，分别刻画了在固定围长和固定悬挂点数条件下具有最大 $A_α$-谱半径的图。
This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are investigated. In particular, we characterize all extremal simple digraphs with the maximum spectral radius among all simple digraphs with arcs number $e=2{k\choose 2}+t$ and $k>4t^4+4$.
Nonregular graphs with a given maximum degree attaining maximum spectral radius
针对具有给定最大度且达到最大谱半径的$n$阶连通非正则图$G$，研究证实了部分度序列猜想并完整刻画了$Δ=n-2$（$n≥5$）和$Δ=n-3$（$n≥59$）时$G$的结构。
Extremal spectral radius of nonregular graphs with a fixed odd maximum degree
本文证明了Liu关于非正则图邻接谱半径与最大度之差的极限猜想在固定奇最大度$\Delta\ge5$时成立，补充了已有$\Delta=3,4$的结果。
Asymptotic maximum spectral radius of nonregular graphs
本文利用有效电阻界、比较图与Wirtinger不等式，证明了连通非正则图最大邻接谱半径的渐近公式，并验证了Liu提出的猜想。
In $1967$, Vizing determined the maximum size of a graph with given order and radius. In $1973$, Fridman answered the same question for digraphs with given order and outradius. We investigate that question when restricting to biconnected digraphs. Biconnected digraphs are the digraphs with a finite total distance and hence the interesting ones, as we want to note a connection between minimizing the total distance and maximizing the size under the same constraints. We characterize the extremal digraphs maximizing the size among all biconnected digraphs of order $n$ and outradius $3$, as well as…
The maximum spectral radius of planner graphs without the joint of K2 and a linear forest
该研究针对不含K₂与线性森林联图的n顶点平面图的最大谱半径问题，确定了三类线性森林对应的极图唯一为2K₁+C_{n-2}，并刻画了含P₄的线性森林对应的极图结构。
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。
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…
For $r\geq 2$ and $p\geq 1$, the $p$-spectral radius of an $r$-uniform hypergraph $H=(V,E)$ on $n$ vertices is defined to be $$ρ_p(H)=\max_{{\bf x}\in \mathbb{R}^n: \|{\bf x}\|_p=1}r \cdot \!\!\!\! \sum_{\{i_1,i_2,\ldots, i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r},$$ where the maximum is taken over all ${\bf x\in \mathbb{R}^n}$ with the $p$-norm equals 1. In this paper, we proved for any integer $r\geq 2$, and any real $p\geq 1$, and any $r$-uniform hypergraph $H$ with $m={s\choose r}$ edges (for some real $s\geq r-1$), we have $$λ_p(H)\leq \frac{rm}{s^{r/p}}.$$ The equality holds if and only…
Extremal results on $k$-stepwise irregular graphs
本文研究了 $k$-步不规则图的极值性质，证明了其存在性并界定了最大度数和图大小，引入了度复杂度的概念。
Two classical concepts of centrality in a graph are the median and the center. The connected notions of the status and the radius of a graph seem to be in no relation. In this paper, however, we show a clear connection of both concepts, as they obtain their minimum and maximum values at the same type of tree graphs. Trees with fixed maximum degree and extremum radius and status, resp., are characterized. The bounds on radius and status can be transferred to general connected graphs via spanning trees. A new method of proof allows not only to regain results of Lin et al. on graphs with extremum…
Nonregular graphs of odd maximum degree with maximum spectral radius
本文证明了奇数最大度非正则图谱半径的两个猜想：极限公式和极图度序列形式，并给出了精确的渐近展开。
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 $δ=Δ-2$…
The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size
本文针对Li等人提出的刻画不含$θ_{1,3,3}$的图中最大谱半径图的问题，确定了给定边数$m$的不含$θ_{1,3,3}$图的最大谱半径并刻画极图，补全了$\mathcal{G}(m,θ_{1,p,q})$（$q\geq p\geq2$）的最大谱半径极图结论。
Signless Laplacian spectral radius of simplicial complexes without $r$-dimensional wheels
本文研究了不含 $r$-维轮的 $n$ 顶点 $r$-维纯单纯复形的最大无符号拉普拉斯谱半径，对足够大的 $n$，确定了极值复形。结果推广了图上无符号拉普拉斯谱的极值结果，并为 Sós、Erdős 和 Brown 的定理提供了谱类比。
The normalized distance Laplacian of a graph $G$ is defined as $\mathcal{D}^\mathcal{L}(G)=T(G)^{-1/2}(T(G)-\mathcal{D}(G))T(G)^{-1/2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjectu…
Extremal graphs for the maximum $A_α$-spectral radius of graphs with order and size
本文在给定顶点数和边数满足特定范围的条件下，完整刻画了具有最大 $A_α$-谱半径的所有极图，并解决了 Chang 和 Tam 提出的相关问题。
Metaheuristics for finding threshold graphs with maximum spectral radius
针对给定顶点数和边数的连通图中寻找最大谱半径图的问题，提出基于变邻域搜索（VNS）和蜂群优化（BCO）的元启发式方法，通过不完整枚举阈值图实现高效求解，初步实验表明VNS更具优势。
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