A note proving the nullity of block graphs is unbounded

Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a $K_2$-free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of $K_2$-free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.