First, a new approach to treat the strongly correlated conjugated-circuit model on two-dimensional networks is made with computational effort comparable to that for corresponding tight-binding models. Toward this end a translationally symmetric arrow assignment is used to construct an antisymmetrically signed "adjacency" matrix for two-dimensional networks. Then symmetry blocking is used to manipulate this "adjacency" matrix, and make the associated conjugated-circuit computations. Second, a series of two-dimensional translationally symmetric structures related to graphite is constructed by means of a kind of local rearrangement on the graphite lattice. A consequent detailed description of rr-electron resonance energy via conjugated-circuit computations is presented for these novel two-dimensional nets as well as several other regular and semiregular nets with vertices of degree 3. Approximate energy estimates indicate that resonance stability depends dominantly on the local topology of the networks, and in particular on the fraction of faces which are hexagonal.