Summary: A graph \(G\) is said to be singular if its adjacency matrix is singular; otherwise it is said to be non-singular. In this paper, we introduce a class of graphs called looped-trees, and find the determinant and the nonsingularity of looped-trees. Moreover, we determine the singularity or non-singularity of the complement of a certain class of trees with diameter 5 by using the results for looped-trees.