Generalized spectral characterization of signed trees

It is well known that the spectra of graphs encode a lot of combinatorial information about the given graphs. A major unsolved question in spectral graph theory is: “What kinds of graphs are determined (up to isomorphism) by their spectrum (DS for short)?”. The problem originates from chemistry and was raised in 1956 by Günthard and Primas [1], which relates Hückle’s theory in chemistry to graph spectra. The above problem is also closely related to a famous problem of Kac [12]: “Can one hear the shape of a drum?” Fisher [11] modelled the drum by a graph, and the frequency of the sound was characterized by the eigenvalues of the graph. Hence, the two problems are essentially the same.

It was commonly believed that every graph is DS until the first counterexample (a pair of cospectral but non-isomorphic trees) was found by Collatz and Sinogowitz [2] in 1957. Another famous result on cospectral graphs was given by Schwenk [16], which states that almost every tree is not DS. For more constructions of cospectral graphs, see, e.g., [10, 13, 17]. However, it turns out that showing a given graph to be DS is generally very hard and challenging. Up to now, only a few graphs with very special structures are known to be DS. We refer the reader to [7, 8] for more background and known results.

In recent years, Wang and Xu [19] and Wang [20, 21] considered a variant of the above problem. For a simple graph $G$, they defined the generalized spectrum of $G$ as the spectrum of $G$ together with that of its complement $\bar{G}$. A graph $G$ is said to be determined by its generalized spectrum (DGS for short), if any graph having the same generalized spectrum as $G$ is necessarily isomorphic to $G$.

Let $G$ be a graph on $n$ vertices with adjacency matrix $A=A(G)$. The walk-matrix of $G$ is defined as

where $e$ is the all-one vector. Wang [20, 21] proved the following theorem.

The problem of spectral determination of ordinary graphs naturally extends to signed graphs. This paper is a continuation along this line of research for signed graphs in the flavour of Theorem 1.1.

Let $\Delta(T)$ be the discriminant of a tree $T$ (see Section 4 for the definition). The main result of the paper is the following theorem.

As an immediately consequence of Theorem 1.2, we have

Theorem 1.2 shows that whenever the underlying tree $T$ with $n$ vertices satisfies a simple arithmetic condition, then all the $2^{n-1}$ signed trees (including $T$ itself) whose underlying is $T$ is DGS. That is, the DGS property of all these signed trees only depends on the underlying graph $T$. This is somewhat unexpected, since given a pair of trees $T$ and $T^{\prime}$, it seems time consuming even to check whether there exist two signed trees with underlying graphs $T$ and $T^{\prime}$ respectively that are generalized cospectral; see Example 1.

We mention that Theorem 1.2 is the best possible in the sense that it is no longer true if $2^{-\frac{n}{2}}\sqrt{\Delta(T)}$ has a multiple odd prime factor. Moreover, the irreducibility assumption of the characteristic polynomial of the tree is essential which cannot be removed; see Remarks 1 and 2 in Section 4.

The rest of the paper is organized as follows. In Section 2, we give some preliminary results that will be needed in the proof of Theorem 1.2. In Section 3, we give a structure theorem, which plays a key role in the paper. In Section 4, we present the proof of Theorem 1.2. Conclusions and future work are given in Section 5.

For the convenience of the reader, we give some preliminary results that will be needed later in the paper. For more results in spectral graphs theory, we refer to [4, 6]

Let $G=(V,E)$ be a simple graph. A signed graph is a graph obtained from $G$ by assigning a sign $1$ or $-1$ to every edge according to a mapping $\sigma:E\rightarrow\{1,-1\}$. We use $\Gamma=(G,\sigma)$ to denote a signed graph with underlying graph $G$ and sign function (signature) $\sigma$. We call a signed graph a signed bipartite graph, if its underlying graph is bipartite.

Let $U$ be a subset of $V$ such that $(U,V\setminus U)$ is a partition of $V$. A switching w.r.t. $U$ (or $V\setminus U$) is an operation that changes all the signs of edges between $U$ and $V\setminus U$, while keeps the others unchanged. Two signed graphs $\Gamma$ and $\Gamma^{\prime}$ are switching-equivalent if $\Gamma^{\prime}$ can be obtained from $\Gamma$ by a switching operation, or equivalently, there exists a diagonal matrix $D$ with all diagonal entry $\pm 1$ such that $DA(\Gamma)D=A(\Gamma^{\prime})$. A signed graph is balanced if every cycle contains an even number of edges with sign -1. It is well-known that a signed graph is balanced if and only it is switching equivalent to a unsigned graph.

Let $\Gamma$ be a signed graph with adjacency matrix $A(\Gamma)$. The characteristic polynomial of $\Gamma$ is defined as the characteristic polynomial of $A(\Gamma)$, i.e., $\phi(\Gamma;x)=\det(xI-A(\Gamma))$, where $I$ is the identity matrix. Two signed graphs $\Gamma$ and $\Gamma^{\prime}$ with adjacency matrices $A(\Gamma)$ and $A(\Gamma^{\prime})$ respectively are called generalized cospectral if

where $J$ is the all-one matrix and $J-I-A(\Gamma)$ formally denotes the complement of $\Gamma$ (it is indeed the complement of $\Gamma$ if every edge of $\Gamma$ has been assigned a positive sign +1). A signed graph $\Gamma$ is said to be determined by the generalized spectrum (DGS for short), if any signed graph that is generalized cospectral with $\Gamma$ is isomorphic to $\Gamma$.

A polynomial $f(x)\in{\mathbb{Q}[x]}$ is irreducible if it cannot be factored into two polynomials with rational coefficients of lower degree. Let $f(x)\in{\mathbb{Q}[x]}$ be an irreducible polynomial with degree $n$ and $\alpha$ be one of its root. Then $\mathbb{Q}(\alpha)=\{c_{0}+c_{1}\alpha+\cdots+c_{n-1}\alpha^{n-1}:c_{i}\in{\mathbb{Q}},~{}0\leq i\leq n-1\}$ is a number field which is isomorphic to $\mathbb{Q}[x]/(f(x))$ and is obtained by adding $\alpha$ to $\mathbb{Q}$; see e.g. [9].

An orthogonal matrix $Q$ is a square matrix such that $Q^{\textup{T}}Q=I_{n}$. It is called rational if every entry of $Q$ is a rational number, and regular if each row sum of $Q$ is $1$, i.e., $Qe=e$, where $e$ is the all-one column vector. Denote by RO${}_{n}(\mathbb{Q})$ the set of all $n$ by $n$ regular orthogonal matrices with rational entries.

In 2006, Wang and Xu [19] initiated the study of the generalized spectral characterization of graphs. For two generalized cospectral graphs $G$ and $H$, they obtained the following result (see also [5]), which plays a fundamental role in their method.

A graph $G$ with $\det W(G)\neq 0$ is called controllable (see [14]), denoted by $\mathcal{G}_{n}$ the set of all controllable graphs on $n$ vertices. For a graph $G\in\mathcal{G}_{n}$, define

Then according to Theorem 2.1, it is easy to obtain the following

The above theorems extend naturally to signed graphs. By Theorem 2.2, finding out the possible structure of all $Q\in\mathcal{Q}(G)$ is a key to determine whether a (signed) graph $G$ is DGS.

The key observation of this paper is the following theorem which shows that for two generalized cospectral signed bipartite graphs with a common irreducible characteristic polynomial, the regular rational orthogonal matrix carried out the similarity of their adjacency matrices has a special structure.

To give the proof of Theorem 3.1, we need several lemmas below.

Next, we collect some simple facts about the relationships of eigenvalues/eigenvectors between the adjacency matrix $A$ of a signed bipartite graph $\Gamma$ and its bipartite-adjacency matrix $M$.

Now, we present the proof of Theorem 3.1.