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In a graph G, a vertex dominates itself and its neighbors. A subset D ⊆ V (G) is a double dominating set of G if D dominates every vertex of G at least twice. A signed graph Σ = (G, σ) is a graph G together with an assignment σ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset D ⊆ V (Σ) is a double dominating set of Σ if it satisfies the following conditions: (i) D is a double dominating set of G, and (ii) Σ[D : V \ D] is balanced, where Σ[D : V \ D] is the subgraph of Σ induced by the edges of Σ with one end point in D and the other end point in V \ D. The cardinality of a minimum double dominating set of Σ is the double domination number γ ×2 (Σ). In this paper, wegive bounds for the double domination number of signed cubic graphs. We also obtain some bounds on the double domination number of signed generalized Petersen graphs and signed I-graphs.

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For m ≥ 3 and n ≥ 1, the m-cycle book graph B(m, n) consists of n copies of the cycle C m with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed B(m, n) is n + 1, and (b) the chromatic number of a signed B(m, n) is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.

In this paper we find the number of different signatures of P (3, 1), P (5, 1) and P (7, 1) up to switching isomorphism, where P (n, k) denotes the generalised Petersen graph, 2k < n. We also count the number of non-isomorphic signatures on P (2n + 1, 1) of size two for all n ≥ 1, and we conjecture that any signature of P (2n + 1, 1), up to switching, is of size at most n + 1.