Fold thickness of middle graph of some trees / Paul David N. Luna.

Undergraduate Thesis (BSAM)- Applied Mathematics. Camarines Sur Polytechnic Colleges, 2026.

Includes bibliographical references.

The largest number of sequence of foldings that preserves the singularity or non-singularity of a graph G is called the fold thickness of G, denoted by fold(G). This paper investigated the fold thickness of the middle graph of some classes of trees. We show that the middle graph of any tree is singular. Based on this foundational property, the fold thickness for the middle graphs of a path Pn, a star Sk , and the corona product Pn⊙K1 is obtained. For the middle graph of a path,M(Pn) where n ≥ 2, folding pendant vertices yields non-singular graphs, whereas folding specific subdivision vertices preserves singularity, leading to fold(M(Pn)) = 0 for n < 4 and fold(M(Pn)) = 2n − 5 for n ≥ 4. For the middle graph of a star, M(Sk), any 1-fold results in a non-singular graph isomorphic to M(Sk) − v1, implying that fold(M(Sk)) = 0. Finally, for the middle graph of Pn ⊙ K1 where n ≥ 3, a uniform folding sequence can be done by folding subdivision vertices from consecutive pendant edges, yielding a fold thickness of fold(M(Pn ⊙ K1)) = 4n − 6.