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 P d − 1 be the path on d − 1 vertices and K 1 , p be the star of p + 1 vertices. Let p = [ p 1 , p 2 , … , p d − 1 ] such that ∀ i , 1 ⩽ i ⩽ d − 1 , p i . Let C ( p ) be the caterpillar obtained from d − 1 stars K 1 , p i and the path P d − 1 by identifying the root of K 1 , p i with the i-vertex of P d − 1 . For a given n ⩾ 2 ( d − 1 ) , let C = { C ( p ) : ∑ i = 1 , d − 1 p i = n − d + 1 } . In this work, we give the caterpillar in C maximizing the algebraic connectivity.