Line graph eigenvalues and line energy of caterpillars

Abstract The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G . A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ⩾ 3 and n ⩾ 2 ( d - 1 ) . Let p = [ p 1 , p 2 , … , p d - 1 ] with p 1 ⩾ 1 , p 2 ⩾ 1 , … , p d - 1 ⩾ 1 such that p 1 + p 2 + ⋯ + p d - 1 = n - d + 1 . Let C ( p ) be the caterpillar obtained from the stars S p 1 + 1 , S p 2 + 1 , … , S p d - 1 + 1 and the path P d - 1 by identifying the root of S p i + 1 with the i -vertex of P d - 1 . The line graph of C ( p ) , denoted by L ( C ( p ) ) , becomes a sequence of cliques K p 1 + 1 , K p 2 + 2 , … , K p d - 2 + 2 , K p d - 1 + 1 , in this order, such that two consecutive cliques have in common exactly one vertex. In this paper, we characterize the eigenvalues and the energy of L ( C ( p ) ) . Explicit formulas are given for the eigenvalues and the energy of L ( C ( a ) ) where a = [ a , a , … , a ] . Finally, a lower bound and an upper bound for the energy of L ( C ( p ) ) are derived.