Trees with diameter less than 5 and non-singular complement Original Research Article

A graph G is said to be singular if its adjacency matrix is singular; otherwise it is said to be non-singular. Non-singular trees have been completely characterized. Here we investigate the complement of a tree with diameter less than 5 for singularity or non-singularity. It is easy to see that every tree with diameter d ⩽ 2 has a singular complement. We prove that the complement of any tree with diameter 3 is non-singular. We prove also that if T is a tree with diameter 4 and central vertex xo, then the complement T of T is non-singular if and only if T/x0 contains at most one component P2. Given any tree T, we prove that T is singular if T⋎ has at least two components P2 for some vertex v ϵ T.