Yes of course, the *supernova graphs* generalise the star-shaped graphs. They are like the stars, but have "more going on in the middle"; for example their core can be any complete multipartite graph. See "Simply-laced isomonodromy systems, Publ. Math. IHES 116 (2012) no. 1, 1–68" especially section 4 on isospectral integrable systems, and the remark there that our big symplectic vector space generalises the "generalised Moser space" of Adams-Harnad-Hurtubise-Previato. This big symplectic vector space is the space of representations of a doubled quiver whose underlying graph is complete k-partite. E.g. we get the triangle, the square, the tetrahedron and the octahedron as simple examples. The stars appear since a complete bipartite graph with just one node in one of the two parts, is star-shaped. Then there is a general story about gluing "legs" on to the core (complete multipartite) graph.

A typical picture might be as in the linked image: 1, or as in the backdrop/poster here:

https://webusers.imj-prg.fr/~philip.boalch/Mars2017/index.html

Beyond this simply-laced setting, a more general "quiver modularity theorem" (**) was conjectured in appendix C of my 2008 paper "Irregular connections and Kac-Moody root systems" and proved by Hiroe-Yamakawa in their 2014 Adv. Math. paper.

For me the key conceptual breakthrough, that took about 10 years to realise, was the following: in the general linear case the Birkhoff orbits O_B (in my 2001 Adv. Math paper) should be viewed as the large symplectic vector space in the Nakajima construction.

In more detail: I understood the orbits O_B since 1999 (my thesis), and knew from a result of Vergne that they had global Darboux coordinates, but it wasn't until the 2008 paper that I saw the relation to quiver varieties; in fact the first observation was really Exercise 3 in the 2007 paper I wrote for the John McKay volume ("Quivers and difference Painleve equations"), and then I saw that the bipartite case essentially came from the "generalised Moser spaces" of Adams et al (and that this already contained all the stars): this bipartite observation was at the end of my 2007 talk in Princeton, archived recently here:

http://web.archive.org/web/20211224113418/https://webusers.imj-prg.fr/~philip.boalch/files/boalch_2007_IAS%20talk,%20Geometry%20of%20irregular%20connections%20on%20curves.pdf

One reason this took so long was because, for a while, I only wanted to use methods that worked for all reductive groups, not just for GL_n.

There are several things worth mentioning that follow from this link between 2d gauge theory and quiver varieties, such as the global Weyl group, the Lax project (the quivers are like Dynkin diagrams for some [wild] nonabelian Hodge spaces), a new theory of multiplicative quiver varieties, graphical Deligne-Simpson problems,...

Finally, if you know the Atiyah-Gowers discussion about the two cultures of mathematics, solitons versus graph theory, it may be worth pointing out that one thus *can* get somewhere in integrable systems by "sitting in an armchair and trying to understand graphs better"...

(**) i.e. identifying Nakajima quiver varieties with symplectic spaces of matrices of rational one-forms, that we think of as moduli spaces M^* of meromorphic connections on the trivial bundle on the Riemann sphere (or equivalently as meromorphic Higgs bundles).