Explanation of the diagram
The diagram is a loose representation of the functional tendencies of various chords.
- At the center is the tonic and the relative minor chord, which can substitute for it, for example, in a deceptive cadence (V-vi). The parentheses around the vi indicate it as a "substitute" chord but one which does not truly replace the tonic. The asterisk likely points to a footnote not included explaining the vi chord's substitute nature.
- The innermost ring comprises chords that primarily serve dominant (i.e., moving to the tonic) functions.
- The next ring is chords that tend to act as pre-dominants.
- The outer ring is chords that can serve as pre-predominants or which can supplant the tonic (i.e., have at least two common tones), typically as part of a tonic expansion.
The #vio7 chord
Within the context of the diagram, this chord can function in two ways: viio7/vii or c.t.o7/V. "c.t." stands for "common tone", and you can read more about common-tone diminished chords here, Diminished chord constructed over the tonic degree?, and you can find a list of additional SE questions/explanations here.
Outside of SE, Wikipedia mentions common-tone diminished seventh chords, this web page discusses the #vi chord specifically, and Florida State professor Nancy Rogers has posted a handout on the subject.
The #vio7 does not have a unique name in the way that the aug6 and Neapolitan chords do.