No two Jellyfish graphs are L-cospectral and Q-cospectral

关于余谱图ons

On cospectral graphons

本文提出余谱图ons的概念，通过三种等价定义揭示其性质，并展示两个无法被余谱图序列近似的例子。

Characterizing cospectral vertices via isospectral reduction

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symme…

Using twins and scaling to construct cospectral graphs for the normalized Laplacian

The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges has been elusive. In this paper we use basic properties of twins and scaling to show how to construct such graphs. We also give examples of families of graphs which are cospectral with a subgraph for the normalized Laplacian matrix.

Cospectral constructions for several graph matrices using cousin vertices

Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signles…

Cospectral pairs of regular graphs with different connectivity

For vertex and edge connectivity we construct infinitely many pairs of regular graphs with the same spectrum, but with different connectivity.

广义Johnson图和Grassmann图的同谱伴侣

Cospectral mates for generalized Johnson and Grassmann graphs

本文构造了三族无穷多的图，证明广义Johnson图和Grassmann图中的某些图不能由谱唯一确定，并给出了与它们同谱但不同构的图。

Large sets of strongly cospectral vertices in Cayley graphs

Strong cospectrality is an equivalence relation on the set of vertices of a graph that is of importance in the study of quantum state transfer in graphs. We construct families of abelian Cayley graphs in which the number of mutually strongly cospectral vertices can be arbitrarily large.

A construction of cospectral graphs for the normalized Laplacian

We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph. We also consider a related result for swapping out biregular bipartite graphs for the matrix $A+tD$. We produce (exponentially) large families of non-bipartite, non-regular graphs which are mutually cospectral, and also give an example of a graph which is cospectral with its complement but is not self-compl…

Constructions of cospectral graphs with different zero forcing numbers

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing, and provide constructions of infinite families of pairs of cospectral graphs which have different values for these numbers. We explore several methods for obtaining these cospectr…

Cospectral regular graphs with and without a perfect matching

For each $b \geq 5$ we construct a pair of cospectral $b$-regular graphs, where one has a perfect matching and the other one not. This solves a research problem posed by the third author at the 22nd British Combinatorial Conference.

根图广义同谱性的刻画及其在图重构中的应用

A characterization of generalized cospectrality of rooted graphs with applications in graph reconstruction

本文通过矩阵刻画具有广义同谱删顶点子图的根图，并据此给出图可重构性的新条件，证明含几乎可控且具非平凡自同构删点子图的图可重构。

A construction of distance cospectral graphs

The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in their edge counts yet are cospectral with respect to the distance matrix. Further, we identify a subgraph switching behavior which constructs additional distance cospectral graphs. The proofs for both constructions rely on a perturbation of (most of) the distance eigenvectors of one graph to yield the distance eigenvectors of the other.

No Threshold graphs are cospectral

A threshold graph G on n vertices is defined by binary sequence of length n. In this paper we present an explicit formula for computing the characteristic polynomial of a threshold graph from its binary sequence. Applications include obtaining a formula for the determinant of adjacency matrix of a threshold graph and showing that no two nonisomorphic threshold graphs are cospectral.

On adjacency and Laplacian cospectral non-isomorphic signed graphs

Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.

通过展开非二部图构造同谱图

Constructing cospectral graphs by unfolding non-bipartite graphs

本文将二部图的展开操作推广到非二部图，利用自反二部图、半自反二部图和多部图的展开构造邻接矩阵同谱但不同构的图。

On cospectrality of gain graphs

We define $G$-cospectrality of two $G$-gain graphs $(Γ,ψ)$ and $(Γ',ψ')$, proving that it is a switching isomorphism invariant. When $G$ is a finite group, we prove that $G$-cospectrality is equivalent to cospectrality with respect to all unitary representations of $G$. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex $v$ can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph $Γ$ with $n$ vertices and $m$ edges, is equal to the num…

Graphs cospectral with a friendship graph or its complement

Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length 4 or 5, then $G\cong F_n$. Moreover if $G$ is connected and planar then $G\cong F_n$. All but one of connected components of $G$ are isomorphic to $K_2$. The complement $\bar{F_n}$ of the friendship graph is det…

一种构造同谱增益图的切换方法

A switching method for constructing cospectral gain graphs

本文提出了一种构造同谱增益图的新切换方法，将增益图视为带号图的推广，并证明了已有的图同谱方法可作为该结果的推论。

The smallest pair of cospectral cubic graphs with different chromatic indexes

Using an exhaustive search on cubic graphs of order 16, we find a unique cospectral pair with different chromatic indexes. This example indicates that the chromatic index of a regular graph is not characterized by its spectrum, which answers a question recently posed in [O. Etesami, W. H. Haemers, On NP-hard graph properties characterized by the spectrum, Discrete Appl. Math., 285(2020)526-529]. We prove that any orthogonal matrix representing the similarity between the two adjacency matrices of the cospectral pair cannot be rational. This implies that the cospectral pair cannot be obtained us…

具有两个不等于 $\pm 1$ 的特征值的带符号图的同谱性结果

Cospectrality results for signed graphs with two eigenvalues unequal to $\pm 1$

本文研究除至多两个特征值外其余均为 $\pm 1$ 的带符号图的同谱分类问题，刻画了同谱对、由谱确定的图，并完全解决了与完全图二部双重覆盖及友谊图 $F_\ell$ 同谱的带符号图分类。

Connected graphs cospectral with a Friendship graph

Let $n$ be any positive integer, the friendship graph $F_n$ consist of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same eigenvalues. Recently in [http://arxiv.org/pdf/1310.6529v1.pdf] it is proved that if $G$ is any graph cospectral with $F_n$ $(n\neq 16)$, then $G\cong F_n$. In this note, we give a proof of special case of the latter: Any connected graph cospectral with $F_n$ is isomorphic to $F_n$. Our proof is independent of ones given in [http://arxiv.org/pdf/1310.6529v1.pdf] and …

由块相似矩阵产生的图的保同谱合并集

Coalescing sets preserving cospectrality of graphs arising from block similarity matrices

本文研究图的合并操作何时保持同谱性，通过相似矩阵的块结构给出充分条件，并覆盖邻接、拉普拉斯、无符号拉普拉斯、距离及广义距离矩阵等类型。

由若干图的联运算构造非正则$A_α$-同谱图

Construction of non-regular $A_α$-cospectral graphs from some join of graphs

该研究在图论同谱问题背景下，计算多种图联运算的$A_α$-特征多项式及正则图情形下的$A_α$-谱，最终构造出非正则的不同构$A_α$-同谱图，丰富了同谱图的构造方法。

二级水平的切换方法用于构造共谱图

Switching methods of level 2 for the construction of cospectral graphs

本文提出两种新的切换方法，并对现有二级水平切换操作进行组合和几何改写，同时引入可约性概念，分类所有不可约切换方法，扩展了先前结果。

几乎所有余图都有同谱伴侣

Almost all cographs have a cospectral mate

本文结合Johnson-Newman广义同谱定理与树的渐近计数工具，证明了几乎所有余图都存在同谱伴侣，从而将Schwenk关于树的经典结论推广到余图。

共谱图的新算术不变量

New arithmetic invariants for cospectral graphs

本文为共谱图建立三个新算术不变量，证明共谱图邻接矩阵幂次与全1向量乘积模4同余，应用于解决猜想、完美匹配/三角形存在性及多项式重构问题，揭示谱性质与组合结构联系。

通过删边得到的同谱图

Cospectral graphs obtained by edge deletion

本文证明在1-游走正则图中删边（包括团内删边）可构造关于邻接、Laplacian、无符号Laplacian和归一化Laplacian矩阵同谱的非同构图族，并推广到Laplacian及无符号Laplacian同谱情形。

Graphs that are cospectral for the distance Laplacian

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. We establish several general methods for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. We establi…

通过子图的轨道构造共谱顶点

Constructing Cospectral Vertices Through Orbits of Subgraphs

本文提出了一种通过子图轨道构造无向图中共谱顶点的方法，并推广到图拉普拉斯矩阵的情形。

图的邻接算子的谱重数函数与同谱无限图

Spectral multiplicity functions of adjacency operators of graphs and cospectral infinite graphs

本文综述无限图邻接算子谱重数函数的已知实例，并构造同谱无限连通图及由谱唯一确定的无限连通图的显式例子。

Enumeration of cospectral and coinvariant graphs

We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The present data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. Finally, w…

与广义菠萝图同谱的图

The graphs which are cospectral with the generalized pineapple graph

本文通过邻接矩阵特征值分析，完整刻画了所有与广义菠萝图 $K_{p,k}^{q}$ 同谱的图，并将 Haemers 等人关于 $K_{p,1}^{q}$ 的结论推广到更一般的情形。

关于同谱非同构二部图的构造

On the construction of cospectral nonisomorphic bipartite graphs

该研究利用分块张量积构造邻接与归一化拉普拉斯矩阵均同谱的二部图，推广了已有构造方法，简化了经典结论证明并拓展了同构性刻画方式。

计数通过切换操作得到的同谱图

Counting cospectral graphs obtained via switching

本研究提出一种通用计数框架，基于新计数方法统计所有已知邻接矩阵切换方法生成的非同构同谱图，推导渐近结果并补充10顶点以内图的计算机枚举结果，拓展了谱图论相关研究。

On the Wiener index, distance cospectrality and transmission regular graphs

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral graphs with different diameter and different Wiener index. A graph is $k$-transmission-regular if its distance matrix has constant row sum equal to $k$. We establish tight upper and lower bounds for the row sum of a $k$-transmission-regular graph in terms of the number of vertices of the graph. Finally, we determine the Wiener index and its complexity for l…

Constructing cospectral signed graphs

A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported …

Fractional revival on non-cospectral vertices

Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices $a$ and $b$ in a graph, there cannot be perfect state transfer from $a$ to any other vertex. No examples of …

Signed graphs cospectral with the path

A signed graph $Γ$ is said to be determined by its spectrum if every signed graph with the same spectrum as $Γ$ is switching isomorphic with $Γ$. Here it is proved that the path $P_n$, interpreted as a signed graph, is determined by its spectrum if and only if $n\equiv 0, 1$, or 2 (mod 4), unless $n\in\{8, 13, 14, 17, 29\}$, or $n=3$.

Construction of cospectral graphs

Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory specially for large graphs. In this paper, we establish that graph theoretical partial transpose of a graph is a potential tool to create non-isomorphic cospectral graphs by considering a graph as a clustered graph.

Non-Geometric Cospectral Mates of Line Graphs with a Linear Representation

For an incidence geometry $\mathcal{G} = (\mathcal{P}, \mathcal{L}, \text{I})$ with a linear representation $\mathcal{T}_n^*(\mathcal{K})$, we apply WQH switching to construct a non-geometric graph $Γ'$ cospectral with the line graph $Γ$ of $\mathcal{G}$. As an application, we show that for $h \geq 2$ and $0 < m < h$, there are strongly regular graphs with parameters $(v, k, λ, μ) = (2^{2h} (2^{m+h}+2^m-2^h), 2^h (2^h+1)(2^m-1), 2^h (2^{m+1}-3), 2^h (2^m-1))$ which are not point graphs of partial geometries of order $(s,t,α) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1)$.

Godsil-McKay switchings for gain graphs

We introduce a switching operation, inspired by the Godsil-McKay switching, in order to obtain pairs of $G$-cospectral gain graphs, that are gain graphs cospectral with respect to every representation of the gain group $G$. For instance, for two signed graphs, this notion of cospectrality is equivalent to the cospectrality of their signed adjacency matrices together with the cospectrality of their underlying graphs. Moreover, we introduce another more flexible switching in order to obtain pairs of gain graphs cospectral with respect to some fixed unitary representation. Many existing notions o…

Constructing Non-isomorphic Signless Laplacian Cospectral Graphs

In this article, we generate large families of non-isomorphic and signless Lalacian cospectral graphs using partial transpose on graphs. Our constructions are significantly powerful. More than $70\%$ of non-isomorphic signless-Laplacian cospectral graphs can be generated with partial transpose when number of vertices is $\le 8$. We have also produced numerous examples of non-isomorphic signless Laplacian cospectral graphs.

关于秩为4的链图的广义同谱伴侣

On the generalized cospectral mates of chain graphs of rank 4

本文研究秩为4的链图的广义谱确定性，推导其广义同谱性的充要条件，证明双星图为DGS，并构造非双星的DGS但非邻接谱确定的无限族。

通过谱、Smith标准形与补图区分图

Distinguishing graphs by their spectra, Smith normal forms and complements

本文研究利用谱和Smith标准形区分图，给出广义余谱图的新刻画，证明Smith标准形可唯一确定星图，并枚举了10个顶点以内图的相关数量结果。

所有与双星 $P_2(1,n)$ 同谱的图的刻画

A characterization of all graphs cospectral to the double star $P_2(1,n)$

本文研究直径为三的双星图的邻接同谱图，构造了与 $P_2(1,2k)$ 同谱的图，并证明奇数 $n$ 时 $P_2(1,n)$ 由其谱确定。

关于 Seidel 矩阵的同谱与积分链图的简短注记

A short note on cospectral and integral chain graphs for Seidel matrix

本文研究了 Seidel 矩阵的同谱与积分链图，推导出生成无限多个同谱不等价链图的公式，并得到一族 Seidel 积分链图，反证否定了 Greaves 提出的关于含三个有理特征值的 Seidel 矩阵是否包含正则图的猜想。

On Hosoya's dormants and sprouts

In a recent series of papers, Hosoya drew the attention to a particular aspect of constructing cospectral graphs by using coalescences: that cospectral graphs can be constructed by attaching multiple copies of a rooted graph in different ways to subsets of vertices of an underlying graph. Our principal focus is to address the expectations and questions raised in Hosoya's papers with regards to this construction. We give an explicit formula for the characteristic polynomial of such multiple coalescences, from which we obtain a necessary and sufficient condition for their cospectrality. We enume…

Strongly Cospectral Vertices

Two vertices $a$ and $b$ in a graph $X$ are cospectral if the vertex-deleted subgraphs $X\setminus a$ and $X\setminus b$ have the same characteristic polynomial. In this paper we investigate a strengthening of this relation on vertices, that arises in investigations of continuous quantum walks. Suppose the vectors $e_a$ for $a$ in $V(X)$ are the standard basis for $\mathbb{R}^{V(X)}$. We say that $a$ and $b$ are strongly cospectral if, for each eigenspace $U$ of $A(X)$, the orthogonal projections of $e_a$ and $e_b$ are either equal or differ only in sign. We develop the basic theory of this co…

Graphs cospectral with NU$(n + 1, q^2)$, $n \ne 3$

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n, q^2)$, $n \ge 2$. Let NU$(n+1, q^2)$ be the graph whose vertices are the points of $PG(n, q^2) \setminus H(n, q^2)$ and two vertices $P_1$, $P_2$ are adjacent if the line joining $P_1$ and $P_2$ is tangent to $H(n, q^2)$. Then NU$(n + 1, q^2)$ is a strongly regular graph. In this paper we show that NU$(n + 1, q^2)$, $n \ne 3$, is not determined by its spectrum.

几乎所有图的高度相对其阶数较小时不具有谱半径伴侣

Almost all graphs have no cospectral mates with height relative small to its order

本文证明几乎所有阶数为n的图的高度为o((n/ln n)^{1/10})时没有谱伴侣，改进了关于固定层级谱伴侣的结果，并涵盖了通过GM切换获得的小块谱图。

Graphs with at most one generalized cospectral mate

Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and $W=[e,Ae,\ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author showed that any graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-\lfloor n/2 \rfloor}\det W$ is odd and square-free. In this paper, we introduce a large family of graphs $\mathcal{F}_n=\{$ $n$-vertex graphs $G\colon\, 2^{-\lfloor n/2 \rfloor}\det W =p^2b$ and rank$W=n-1$ over $\mathbb{Z}/p\mathbb{Z}\},$ where $b$ is odd and square-free, $p$ is an o…

利用若干corona运算构造$A_α$-同谱图

Constructions of $A_α$-cospectral graphs using some corona operations

本文针对正则图计算三类corona运算所得图的$A_α$-谱，并据此构造出无穷多对$A_α$-同谱图。

High-ordered spectral characterizations of graphs

The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered cospectral. If all graphs who are high-ordered cospectral with the graph $G$ are isomorphic to $G$, we say that $G$ is determined by the high-ordered spectrum.In this paper, we use the high-ordered spectrum of graphs to study graph isomorphism and show that all Smith's graphs are determined by the high-ordered spectrum.And we give infinitely many pairs of trees wi…

通过二级和三级正则有理正交矩阵构造同谱图

Constructing cospectral graphs via regular rational orthogonal matrix with level two and three

该研究针对同谱图构造问题，提出利用二级和三级正则有理正交矩阵的构造思路，给出两种刻画邻接矩阵性质的算法，并提出两种新的切换方法，在一定程度上推广了经典的GM-switching方法。