Constructing signed strongly regular graphs via star complement technique

Abstract

We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notion of signed strongly regular graph is recently defined by the author (Signed strongly regular graphs, Proceeding of 48th Annual Iranian Mathematical Conference, 2017). We construct some families of signed strongly regular graphs with only two distinct eigenvalues. The construction is based on the well-known method known as star complement technique.

Some regular signed graphs with only two distinct eigenvalues

“…The following is a general determination of STE's. The result is independently obtained in [15] and [18]. Recently some problems on the spectrum of signed adjacency matrices have attracted many studies.…”

“…The construction problem of STE's is also mentioned there. There are some results on the problem in literature, see [9,14,15,18]. It is known that the only graphs with two distinct eigenvalues of the ordinary adjacency matrix are the complete graphs, which is extensively far from the known results for the signed graphs.…”

“…For more details on the notions and definitions see [17,19]. It is well adopted for the family of signed graphs as well as any other symmetric matrices, see [2,15]. During the preparation of this paper we encounter interesting results from other combinatorial areas such as symmetric weighing matrices, Hadamard matrices, Conference matrices and Ramanujan graphs.…”

Unit gain graphs with two distinct eigenvalue and systems of lines in complex space

“…However, contrary to [41], the current context does not guarantee that a is integer. Consider the following example, constructed from an equiangular tight frame of 7 vectors in dimension 3, which is closely related to the Fano plane.…”

“…The smallest possible n is n = 10. Ramezani [41] has constructed a signed graph (gains in T 2 ⊂ T 6 ) with the above spectrum on the complement of the Petersen graph. Ramezani moreover shows that this example, illustrated in Figure 2a, is actually a member of an infinite family of signed graphs with two distinct eigenvalues on the triangular graphs 5 ∆(m).…”