Let $\mathcal A$ and $\mathcal B$ be graph algebras. In this paper we present the notion of an ideal in a graph algebra and prove that an ideal extension of $\mathcal A$ by $\mathcal B$ always exists. We describe (up to isomorphism) all such extensions.
Let D be an oriented graph of order n and size m. A γ-labeling of D is a one-to-one function f : V (D) → {0, 1, 2, . . . , m} that induces a labeling f ' : E(D) → {±1, ±2, . . . , ±m} of the arcs of D defined by f ' (e) = f(v) − f(u) for each arc e = (u, v) of D. The value of a γ-labeling f is val(f) = ∑ e∈E(G) f ' (e). A γ-labeling of D is balanced if the value of f is 0. An oriented graph D is balanced if D has a balanced labeling. A graph G is orientably balanced if G has a balanced orientation. It is shown that a connected graph G of order n ≥ 2 is orientably balanced unless G is a tree, n ≡ 2 (mod 4), and every vertex of G has odd degree.