Median eigenvalues of subcubic graphs

In mathematical chemistry, graphs where each vertex connects
to at most three others are often referred to as chemical graphs. In
2010, Fowler and Pisanski conjectured that, with only a finite number
of exceptions, the median eigenvalues of chemical graphs lie in the
interval [-1,1]. They confirmed this for all chemical trees. Building
upon this, Mohar in 2013 extended the validation to all bipartite
planar chemical graphs and, later in 2016, to all bipartite chemical
graphs except the Heawood graph. In this talk we will look at our
recent work where we fully resolve this conjecture by proving that,
except the Heawood graph, the median eigenvalues of all connected
chemical graphs lie within [-1,1]. Additionally, we will see that a
positive fraction of the eigenvalues around the median eigenvalues
also reside in this interval, mirroring Mohar’s findings for bipartite
chemical graphs. This is talk is based on joint work with Dr. Zilin
Jiang and Benjamin Jeter in arXiv 2502.13139