Some regular signed graphs with only two distinct eigenvalues

Abstract

We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the {5, 6, . . . , 10}-regular signed graphs which have only two distinct eigenvalues. For each obtained parameter we provide some examples of signed graphs having two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of each mentioned valency with only two distinct eigenvalues. We prove that for any k ≥ 5 there are infinitely many connected signed k-regular graphs having maximum eigenvalue √ k. Moreover for each m ≥ 4 we construct a signed 8-regular graph with spectrum [4 m , −2 2m ]. These yield infinite family of k-regular Ramanujan graphs, for each k.

More on signed graphs with at most three eigenvalues

“…This theorem enables us to construct an infinite family of signed graphs with spectrum − √ α n+m , √ α n−m , 2 √ α m for some appropriate α. We remark that the matrix (11) does not always correspond to a signed graph. For a signed graph, the inner product of any row of W 1 and any row of W 2 has to be 0 or ± √ α, and in [11] one can find a method for constructing a family of weighing matrices of weight 4 with this property.…”

“…We remark that the matrix (11) does not always correspond to a signed graph. For a signed graph, the inner product of any row of W 1 and any row of W 2 has to be 0 or ± √ α, and in [11] one can find a method for constructing a family of weighing matrices of weight 4 with this property. The method can be used to construct signed graphs with spectrum [−2 n+m , 2 n−m , 4 m ], as in the following example.…”

Open problems in the spectral theory of signed graphs

“…In the above category we find the complete graphs with homogeneous signatures (K n , +) and (K n , −), the maximal cyclotomic signed graphs T 2k , S 14 and S 16 , and that list is not complete (for example, the unbalanced 4-cycle C − 4 and the 3-dimensional cube whose cycles are all negative must be included). There is already some literature on this problem, and we refer the readers to see [37,71]. All such graphs have in common the property that positive and negative walks of length greater than or equal to 2 between two different and non-adjacent vertices are equal in number.…”

Induced subgraphs of product graphs and a generalization of Huang's theorem

“…For basic results in the theory of signed graphs, the reader is referred to Zaslavsky [24]. Recently, the spectra of signed graphs have attracted much attention, as found in [1,2,4,6,8,9,14,18,19,22,25], among others. In [2], the authors surveyed some general results on the adjacency spectra of signed graphs and proposed some spectral problems which are inspired by the spectral theory of unsigned graphs.…”

“…In [2], the authors surveyed some general results on the adjacency spectra of signed graphs and proposed some spectral problems which are inspired by the spectral theory of unsigned graphs. In particular, the signed graphs with exactly two distinct eigenvalues have been greatly investigated in recent years, see [8,14,16,18,19,22]. In [14], Hou, Tang, and Wang characterized all simple connected signed graphs with maximum degree at most 4 and with just two distinct adjacency eigenvalues.…”