Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ : E(G) −→ {−, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if Γ is a signed complete graph of order n with k negative edges, k < n − 1 and Γ has maximum index, then negative edges form K 1,k . In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph. We show that the eigenvalues of Γ satisfy the following inequalities:−5 ≤ λ n ≤ · · · ≤ λ 2 ≤ 3.

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