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 to signed graphs, and we can build pairs of cospectral switching nonisomorphic 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.
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 $n=3$.
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 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…
A general switching method for constructing E-cospectral hypergraphs
Eindhoven University of Technology（埃因霍温理工大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；University of South Carolina（南卡罗来纳大学）；Technische Universität Bergakademie Freiberg（弗莱贝格矿业工业大学）
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.
We construct infinitely many signed graphs having symmetric spectrum, by using the NEPS and rooted product of signed graphs. We also present a method for constructing large cospectral signed graphs. Although the obtained family contains only a minority of signed graphs, it strengthen the belief that the signed graphs with symmetric spectrum are deeper than bipartite graphs, i.e the unsigned graphs with symmetric spectrum.
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…
The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path $P_n$ with alternating signs has the minimum Wiener index among all signed trees with $n$ vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever $n$ is at least 30.
We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We construct some families of bipartite signed graphs with only two distinct eigenvalues. This leads to constructing infinite families of regular bipartite Ramanujan graphs.
In 1985, Arjeh Cohen and Jacques Tits proved the existence of a 4-cycle-free 2-fold cover of the hypercube. This Cohen-Tits cover is closely related to the signed adjacency matrix that Hao Huang used last year in his proof of the Sensitivity Conjecture. Terence Tao observed that Huang's signed adjacency matrix can be understood by lifting functions on an elementary abelian 2-group to functions on a central extension. Inspired by Tao's observation, we generalize the Cohen-Tits cover by constructing, as Cayley graphs for extraspecial p-groups, two infinite families of 4-cycle-free p-fold covers …
We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. Specific results we prove include: (1) The signed join of a negative 2-clique with any positive (n,3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. (2) A signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for crea…
The notion of an Abstract Young (briefly: AY) representation is a natural generalization of the classical Young orthogonal form. The AY representations of the symmetric group are characterized by Adin, Brenti and Roichman in [U2]. In this paper we present several types of minimal AY representation of $D_n$ associated with standard D-Young tableaux which are a natural generalization of usual standard Young tableaux. We give an explicit combinatorial view (the representation space is spanned by certain standard tableaux while the action is a generalized Young orthogonal form) of representations …
