Supriyo Dutta, Bibhas Adhikari
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.
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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.
Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family…
In this paper, it is proved that the jellyfish graphs, a natural generalization of sun graphs, are both DLS and DQS.
We prove that for a pair of cospectral graphs G and H, there exist their non trivial lifts G0 and H0 which are cospectral. More over for a pair of cospectral graphs on 6 vertices, we find some cospectral lifts of them.
Singularly cospectral circulant graphs
该研究针对循环图的奇异同谱问题，给出偶数顶点循环图为非同谱奇异同谱图的充分条件，分析其惯性异同，并证明奇素数顶点的奇异同谱循环图必同构。
A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.
We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral graphs with differing number of edges, including situations where one graph is a subgraph of the other. The method used to demonstrate cospectrality is by showing the characteristic polynomials are equal.
In [Steve Butler. A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390, 2010.], Butler constructed a family of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. In this article, we extend this construction for generating larger classes of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. Also, we provide a couple of constructions of non-bipartite graphs, which are cospectral for the adjacency matrices but not necessaril…
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。
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…
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.
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…
For vertex and edge connectivity we construct infinitely many pairs of regular graphs with the same spectrum, but with different connectivity.
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。
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.
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…
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…
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.
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。
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.
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.
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.
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。
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…
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。
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…
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。
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。
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 …
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。
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。
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。
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。
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。
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。
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…
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。
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。
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…
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。
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。
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。
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。
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…
Cospectral vertices, walk-regular planar graphs and the echolocation problem
本文研究有限图上的同谱顶点，提出简单判定准则，结合计算机搜索找到最小反例，并分类所有3连通游走正则平面图，证明其必为顶点传递。
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 …
Originating from spectral graph theory, cospectrality is a powerful generalization of exchange symmetry and can be applied to all real-valued symmetric matrices. Two vertices of an undirected graph with real edge weights are cospectral iff the underlying weighted adjacency matrix $M$ fulfills $[M^k]_{u,u} = [M^k]_{v,v}$ for all non-negative integer $k$, and as a result any eigenvector $ϕ$ of $M$ has (or, in the presence of degeneracies, can be chosen to have) definite parity on $u$ and $v$. We here show that the powers of a matrix with cospectral vertices induce further local relations on its …
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 …
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 $n=3$.
On the number of generalized cospectral mates of graphs
本文确定了简单图广义共谱伴侣数量的上限，基于行走矩阵的Smith Normal Form得出紧致上界，扩展了具有强谱唯一性结果的图类。
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 $(s,t,α) = ((2^h+1)(2^m-1), 2^h-1, 2^m-1)$.
Strongly cospectral vertices in normal Cayley graphs
本文证明了正规Cayley图中两两强余谱顶点数量的上界（由特征值重数决定），确定了$\mathbb{Z}_2^d$和$\mathbb{Z}_4^d$中的显式界限，并给出了包含四个两两强余谱顶点的无限族图，证明其各维度均存在。
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…
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。
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。
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。
Walk Matrix-Based Upper Bounds on Generalized Cospectral Mates
本文研究图有多少个非同构广义同谱伴侣，引入一类广泛图，利用游走矩阵行列式的算术结构给出其上界的显式判据。
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。
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…
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…
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