Determination of particular double starlike trees by the Laplacian spectrum

Abstract

A double starlike tree is a tree in which exactly two vertices have degree greater than two. In this study we consider double starlike trees obtained by attaching p−2(forp≥3) pendant vertices at an internal vertex and q−2(q≥3) pendant vertices at a different internal vertex of a fixed path P. We denote this tree by T≅D(a,b,c,p,q), where a,b,c stand for the numbers of vertices in segments of P obtained by deleting vertices of degree p and q. It is known that, depending on parameters, T may or may not be determined by its Laplacian spectrum. In the latter case we provide the structure of a putative tree with the same Laplacian spectrum. This result implies the known result stating that D(1,b,1,p,q) is determined by the Laplacian spectrum and two new results stating the same for D(1,b,2,p,p) and D(2,b,2,p,p).