On determination of caterpillars with four terminal vertices by their Laplacian spectrum

We consider the class of caterpillars with four terminal vertices. Here we prove that every of such caterpillar whose internal path differs in length from both 1 to 3 is uniquely determined by its Laplacian spectrum. Next we take into consideration the remaining two possibilities for the internal path. In the first situation we prove that there is exactly one caterpillar which is not determined by its Laplacian spectrum, while we find an infinite family of such caterpillars in the second. Finally, some observations are given.