Signed graphs with two eigenvalues and vertex degree five

Abstract

It is known that a signed graph with exactly 2 eigenvalues must be regular, and all those whose vertex degree does not exceed 4 are known. In this paper we characterize all signed graphs with 2 eigenvalues and vertex degree 5. We also determine all signed graphs with 2 eigenvalues and 12 or 13 vertices, which is a natural step since those with a fewer number of vertices are known.

Signed (0,2)‐graphs with few eigenvalues and a symmetric spectrum

“…Next, SR2SEs with $$r=5$$ can be deduced from the incomplete list of signed graphs with eigenvalues $$\pm \sqrt{5}$$ of [18]. Namely, this list contains all signed rectagraphs and among them there are exactly four SR2SEs.…”

“…The three of them are bipartite and they are obtained as in Theorem 3.5 on the basis of weighing matrices of order 12, 14, and 16, respectively. These matrices can be found in [9,18] (in both references they are denoted by $${W}_{12,5},{W}_{14,5}$$ and $$D(16,5)$$). The latter one is the smallest nonbipartite SR2SE illustrated in Figure 2B.…”

“…We denote the obtained SR2SEs by R1.1, R2.1, R3.1, R4.1, and R4.2, respectively. Next, SR2SEs with r = 5 can be deduced from the incomplete list of signed graphs with eigenvalues ± 5 of [18]. Namely, this list contains all signed rectagraphs and among them there are exactly four SR2SEs.…”

Signed $(0,2)$-graphs with few eigenvalues and a symmetric spectrum

“…We denote the obtained SR2SEs by R1.1, R2.1, R3.1, R4.1 and R4.2, respectively. Next, SR2SEs with r = 5 can be deduced from the incomplete list of signed graphs with eigenvalues ± √ 5 of [18]. Namely, this list contains all signed rectagraphs and among them there are exactly 4 SR2SEs.…”

“…The three of them are bipartite and they are obtained as in Theorem 3.5 on the basis of weighing matrices of order 12, 14 and 16, respectively. These matrices can be found in [9,18] (in both references they are denoted by W 12,5 , W 14,5 and D (16,5)). The latter one is the smallest non-bipartite SR2SE illustrated in Figure 2(b).…”