Walks and the spectral radius of graphs

On the spectral radius of unicyclic and bicyclic graphs with a fixed diameter

The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.

On the generalized distance spectral radius of graphs

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_α(G)=αTr(G)+(1-α)D(G), \;\;0\leα\le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance matrix and diagonal matrix of the vertex transmissions of $G$, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs w…

Maximal digraphs whose Hermitian spectral radius is at most $2$

We classify maximal digraphs whose Hermitian spectral radius is at most $2$.

Distance spectral radius of uniform hypergraphs

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.

Sufficient conditions for Hamilton-connected graphs in terms of (signless Laplacian) spectral radius

In this paper, we present some spectral sufficient conditions for a graph to be Hamilton-connected in terms of the spectral radius or signless Laplacian spectral radius of the graph. Our results improve some previous work.

The bounds of the spectral radius of general hypergraphs in terms of clique number

The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.

The p-spectral radius of k-partite and k-chromatic uniform hypergraphs

We characterize the r-graph with maximal p-spectral radius among the k-partite r-graphs of order n, and the 3-graph with maximal p-spectral radius among the k-chromatic 3-graphs of order n.

Connected Hypergraphs with Small Spectral Radius

In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $ρ_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with …

Sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor and its applications

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. We also apply these bounds to the adjacency spectral radius and signless Laplacian spectral radius of a uniform hypergraph.

Maxima of the signless Laplacian spectral radius for planar graphs

The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph $K_{2}\nabla P_{n-2}$ has the maximal signless Laplacian spectral radius among all planar graphs of order $n\geq 456$.

On the $α$-spectral radius of uniform hypergraphs

For $0\leα<1$ and a uniform hypergraph $G$, the $α$-spectral radius of $G$ is the largest $H$-eigenvalue of $α\mathcal{D}(G) +(1-α)\mathcal{A}(G)$, where $\mathcal{D}(G)$ and $\mathcal{A}(G)$ are the diagonal tensor of degrees and the adjacency tensor of $G$, respectively. We give upper bounds for the $α$-spectral radius of a uniform hypergraph, propose some transformations that increase the $α$-spectral radius, and determine the unique hypergraphs with maximum $α$-spectral radius in some classes of uniform hypergraphs.

关于树的补图的距离谱半径与距离无符号拉普拉斯谱半径的一个注记

A note on the distance and distance signless Laplacian spectral radius of complements of trees

该研究通过分析广义树移位操作对树补图三类谱半径的增大作用，修正了既有相关定理证明中的歧义。

Distance spectral radius of a tree with given diameter

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely characterized.

On some properties of the $α$-spectral radius of the $k$-uniform hypergraph

In this paper we show how the $α$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $α$-spectral radius among $k$-uniform non-caterpillar hypergraphs with given order, size and diameter. we also characerize the second largest $α$-spectral radius among all $k$-uniform supertrees on $n$ vertices by two methods.

不含\(\mathcal{F}\)图类的广义谱封闭性

Generalized spectral closedness of $\mathcal{F}$-free graph classes

研究由禁止导出子图族\(\mathcal{F}\)定义的图类的广义谱封闭性，引入模式闭迹概念，通过建立相关展开式得到代数充分条件，将寻找支撑子图问题归结为线性规划问题，并证明阈值图类和链图类具有广义谱封闭性。

The distance Laplacian spectral radius of unicyclic graphs

For a connected graph $G$, the distance Laplacian spectral radius of $G$ is the spectral radius of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is a diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. In this paper, we determine the unique graphs with maximum distance Laplacian spectral radius among unicyclic graphs.

Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D) = diag(D) + A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give sharp bound on q(D) with outdegree sequence and compare the bound with some known bounds, establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the corresponding extremal …

Principal eigenvector and spectral radius of uniform hypergraphs

In this paper, we give some bounds for principal eigenvector and spectral radius of connected uniform hypergraphs in terms of vertex degrees, the diameter, and the number of vertices and edges.

The linear unicyclic hypergraph with the second or third largest spectral radius

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

论谱半径序的有限性

On finiteness of spectral radius order

本文针对谱半径序难以显式计算和刻画有限性的问题，刻画了二次代数整数及不大于2的数的有限谱半径序，并推导了两类无限二次代数整数族的谱半径序精确值。

Maximum spectral radius of graphs with given connectivity and minimum degree

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_{…

The non-bipartite integral graphs with spectral radius three

In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.

The spectral radius of subgraphs of regular graphs

We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter.

On the spectral radius of simple digraphs with prescribed number of arcs

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 $k>4t^4+4$.

Spectral radius and the $2$-power of Hamilton cycles

Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $μ(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without containing the $2$-power of a Hamilton cycle.

A proof of a conjecture on the distance spectral radius

A cactus is a connected graph in which any two cycles have at most one common vertex. We determine the unique graph that maximizes the distance spectral radius over all cacti with fixed numbers of vertices and cycles, and thus prove a conjecture on the distance spectral radius of cacti in [S.S. Bose, M. Nath, S. Paul, On the distance spectral radius of cacti, Linear Algebra Appl. 437 (2012) 2128--2141]. We prove the result in the context of hypertrees.

超树谱半径的代数重数

The algebraic multiplicity of the spectral radius of a hypertree

本文利用Poisson公式和匹配多项式，解决了连通一致超图谱半径代数重数未知的问题，确定了一致超树谱半径的代数重数。

The Dimension of Eigenvariety of Nonnegative Tensors Associated with Spectral Radius

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this paper we prove that the dimension of the above projective eigenvariety is zero, i.e. there are finite many eigenvectors associated with t…

树的谱游走判定的一个严格匹配数阈值

A Sharp Matching-Number Threshold for Spectral-Walk Determination of Trees

本文确定树的谱游走判定的匹配数阈值：匹配数≤4的树可由邻接谱与3≤k≤8的总游走序列唯一确定，匹配数为5时存在反例，证明基于有限核约化与代数重构。

On the maximal $α$-spectral radius of graphs with given matching number

Let $\mathscr{G}_{n,β}$ be the set of graphs of order $n$ with given matching number $β$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_α(G)=αD(G)+A(G)$ is called the $α$-spectral radius of $G$. The graphs with maximal $α$-spectral radius in $\mathscr{G}_{n,β}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $\mathscr{G}_{n,β}$. More precisely, we generalize the known results on t…

双环图和二分单环图中极值ABS谱半径

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。

Minimum Distance Spectral Radius of Graphs with Given Edge Connectivity

In this paper we determine the unique graph with minimum distance spectral radius among all connected graphs of fixed order and given edge connectivity.

The largest spectral radius of uniform hypertrees with a given size of matching

In this paper, using the theory of matching polynomial of hypertrees and ordering of hypertrees, we determine the largest spectral radius of hypertrees with $m$ edges and given size of matching.

The maximum spectral radius of C_4-free graphs of given order and size

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.

The effect on the spectral radius of r-graphs by grafting or contracting edges

Let $\mathcal{H}^{(r)}_n$ be the set of all connected $r$-graphs with given size $n$. In this paper, we investigate the effect on the spectral radius of $r$-uniform hypergraphs by grafting or contracting an edge and then give the ordering of the $r$-graphs with small spectral radius over $\mathcal{H}^{(r)}_n$, when $n\geq 20$.

Maximal distance spectral radius of 4-chromatic planar graphs

We show that the kite graph $K_4^{(n)}$ uniquely maximizes the distance spectral radius among all connected $4$-chromatic planar graphs on $n$ vertices.

Maxima of spectral radius of irregular graphs with given maximum degree

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 …

关于平面图的最大谱半径

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。

Spectral Radius and Hamiltonicity of graphs

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasi-complement respectively.

The spectral radius of the square of graphs

The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and unicyclic graphs respectively.We also investigate the spectral radius of the square of unicyclic graphs with given girth or trees with fixed diameter.

Spectral radius of uniform hypergraphs and degree sequences

In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution $U$ by lifting the eigenvalues of an induced self-adjoint matrix $T$ onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulab…

加权有向图上非回溯游走的生成函数：收敛半径与Ihara定理

Generating functions of non-backtracking walks on weighted digraphs: radius of convergence and Ihara's theorem

本文利用多项式与有理矩阵理论，研究了一般有向和/或加权图上非回溯游走生成函数的收敛半径，给出了有向无权图的刻画及加权图的精确公式，并证明了回溯降权游走下的Ihara定理。

块图补图的谱半径与距离谱半径

The spectral radius and the distance spectral radius of complements of block graphs

本文确定了在团树补图和块图补图中，谱半径与距离谱半径达到极值的图。

Erdös-Rényi 随机图中大相互作用半径下行走和三角形的极限定理

Limit theorems for walks and triangles on Erdös-Rényi random graphs with large interaction radius

本文研究了Erdös-Rényi随机图中q步行走数量的累积量，在顶点数N、浓度c和相互作用半径R趋于无穷时，通过树型图示来关联这些累积量，并证明了非闭合行走和三角形数量的极限定理，同时指出了三角形数量在随机图中的渐近阈值。

通过确定性差异游走实现谱稀疏化

Spectral Sparsification by Deterministic Discrepancy Walk

本文将确定性差异游走框架从向量差异推广到矩阵差异，统一处理多种谱稀疏化问题，并以确定性分析给出更优结果与更简证明。

Remarks on random walks on graphs and the Floyd boundary

We show that for a uniformly irreducible random walk on a graph, with bounded range, there is a Floyd function for which the random walk converges to its corresponding Floyd boundary. Moreover if we add the assumptions, $p^{(n)}(v,w)\leq C ρ^n$, where $ρ< 1$ is the spectral radius, then for any Floyd function $f$ that satisfies $\sum_{n=1}^{\infty}nf(n)<\infty$, the Dirichlet problem with respect to the Floyd boundary is solvable.

Cliques and the Spectral Radius

We present a number of relations involving the number of cliques in a graph and its spectral radius.

Bounds via spectral radius-preserving row sum expansions

We show a simple method for constructing larger matrices but preserving the spectral radius. This yields a sufficient criteria for two square matrices of arbitrary dimension have the same spectral radius, a way to compare spectral radii of two matrices, and a way to derive new upper and lower bounds on spectral radius which give the standard row sum bounds as a special case.

On the distance $α$-spectral radius of a connected graph

For a connected graph $G$ and $α\in [0,1)$, the distance $α$-spectral radius of $G$ is the spectral radius of the matrix $D_α(G)$ defined as $D_α(G)=αT(G)+(1-α)D(G)$, where $T(G)$ is a diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. We give bounds for the distance $α$-spectral radius, especially for graphs that are not transmission regular, propose some graft transformations that decrease or increase the distance $α$-spectral radius, and determine the unique graphs with minimum and maximum distance $α$-spectral radius among some classes of graphs.

基于单纯复形上随机游走的超图稀疏割

Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

本研究衔接超图研究的两大分支，分析单纯复形上不同游走的谱隙与超图扩张、电导率的关联，证明部分游走谱隙无法推导超图电导率界，给出类Cheeger不等式，并验证链环扩张无法以类Cheeger方式界超图扩张。

Characterizations of the spectral radius of nonnegative weakly irreducible tensors via digraph

For a nonnegative weakly irreducible tensor $\mathcal{A}$, we give some characterizations of the spectral radius of $\mathcal{A}$, by using the digraph of tensors. As applications, some bounds on the spectral radius of the adjacency tensor and the signless Laplacian tensor of the $k$-uniform hypergraphs are shown.

Hypergraphs with Spectral Radius at most $(r-1)!\sqrt[r]{2+\sqrt{5}}$

In our previous paper, we classified all $r$-uniform hypergraphs with spectral radius at most $(r-1)!\sqrt[r]{4}$, which directly generalizes Smith's theorem for the graph case $r=2$. It is nature to ask the structures of the hypergraphs with spectral radius slightly beyond $(r-1)!\sqrt[r]{4}$. For $r=2$, the graphs with spectral radius at most $\sqrt{2+\sqrt{5}}$ are classified by [{\em Brouwer-Neumaier, Linear Algebra Appl., 1989}]. Here we consider the $r$-uniform hypergraphs $H$ with spectral radius at most $(r-1)!\sqrt[r]{2+\sqrt{5}}$. We show that $H$ must have a quipus-structure, which …

树游走与随机图的谱

Tree walks and the spectrum of random graphs

本研究针对随机图G(n, c/n)谱测度研究不足的问题，结合并拓展两种已有的组合方法，通过计数生成树的游走逼近该类随机图谱测度的矩，丰富了随机图谱理论的相关研究。