On nut and core singular fullerenes

A graph G   is singular of nullity η(>0)η(>0), if its adjacency matrix AA is singular, with the eigenvalue zero of multiplicity ηη. A singular graph having a 0-eigenvector, xx, with no zero entries, is called a core graph. We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the entries of xx which lead to interesting implications in chemistry. The zero eigenvalue is rare in a fullerene graph. We show that there are possible nut fullerenes with relatively simple structures.