[ ] : a term (usually, one rarely used) for which there is a preferred variant.
Citations are to ``A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas'', Electronic Journal of Combinatorics (1998), Dynamic Surveys in Combinatorics #DS8.
Since HTML provides poorly for mathematical notation, for correct notations one should consult the dvi, PostScript, or PDF version.
To simplify descriptions I adopt some standard notation. I generally call a graph Gamma, a signed graph Sigma, a gain graph (and, when indicated by the context, a permutation gain graph) Phi, and a biased graph Omega = (Gamma, B). The sign function of Sigma is sigma, the gain function of Phi is phi; that is, I use upper and lower case consistently for the graph and its edge labelling. The gain group of Phi is G.
These definitions about graphs are intended not to be a glossary of graph theory but to clarify the special usages appropriate to signed, gain, and biased graphs.
The points of the matroid are the edges (except for the extra point in the extended lift). A loose edge is a circuit and a half edge acts like an unbalanced loop.
The molecules of interest seem to be hydrocarbons. The graphs are those of the carbon skeleta. Usage does not seem to be completely consistent.
I take vectors to be row vectors so the orthogonal (or unitary) group acts on the right; this is for consistency with gain calculations.