Maximizing the algebraic connectivity for a subclass of caterpillars

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d⩾3 and n⩾6 be given. Let Pd−1 be the path on d−1 vertices and K1,p be the star of p+1 vertices. Let p=[p1,p2,…,pd−1] such that ∀i,1⩽i⩽d−1,pi. Let C(p) be the caterpillar obtained from d−1 stars K1,pi and the path Pd−1 by identifying the root of K1,pi with the i-vertex of Pd−1. For a given n⩾2(d−1), let C={C(p):∑i=1,d−1pi=n−d+1}. In this work, we give the caterpillar in C maximizing the algebraic connectivity.