We investigate the multiplicity and uniqueness of positive solutions for the superlinear singular (p, q)-Laplacian equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u-\Delta _q u+a(x)u^{p-1}+b(x)u^{q-1}=f(x)u^{-\gamma }+\lambda g(x)u^{\alpha }, \;\;\;\;\text{ in }\;\; V,\\ u>0,\;\;u\in W_a^{1,p}(V) \cap W_b^{1,q}(V), \end{array}\right. } \end{aligned}\) on a weighted locally finite graph \(G=(V,E)\) , where \(0<\gamma<1<q\le p<\alpha +1\) and \(\lambda \) is a parameter. The potential functions a(x) and b(x) satisfy some suitable conditions. The functions f and g satisfy \(f>0, g \ge 0\) with \(f\in L^1(V)\cap L^{\frac{p}{p-1+\gamma }}(V) \cap L^{\frac{q}{q-1+\gamma }}(V)\) and \(g\in L^1(V)\cap L^\infty (V)\) . By making use of the method of Nehari manifold and the Ekeland’s variational principle, we prove that there exist two positive solutions for \(\lambda \) belonging to some precise interval. Besides, we also investigate the existence and uniqueness of positive solution for \(\lambda <0\) . We overcome some difficulties which are caused by: (i) the singular term; (ii) the definition of gradient \(|\nabla u|\) on graph which is different from that on \(\mathbb {R}^N\) ; (iii) the lack of compactness of Sobolev embedding.
