Line graph eigenvalues and line energy of caterpillars

The energy of a graph G   is the sum of the absolute values of the eigenvalues of the adjacency matrix of GG. A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d⩾3d⩾3 and n⩾2(d-1)n⩾2(d-1). Let p=[p1,p2,…,pd-1]p=[p1,p2,…,pd-1] with p1⩾1,p2⩾1,…,pd-1⩾1p1⩾1,p2⩾1,…,pd-1⩾1 such thatp1+p2+⋯+pd-1=n-d+1.p1+p2+⋯+pd-1=n-d+1.Let C(p)C(p) be the caterpillar obtained from the stars Sp1+1,Sp2+1,…,Spd-1+1Sp1+1,Sp2+1,…,Spd-1+1 and the path Pd-1Pd-1 by identifying the root of Spi+1Spi+1 with the ii-vertex of Pd-1Pd-1. The line graph of C(p)C(p), denoted by L(C(p))L(C(p)), becomes a sequence of cliques Kp1+1,Kp2+2,…,Kpd-2+2,Kpd-1+1Kp1+1,Kp2+2,…,Kpd-2+2,Kpd-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))L(C(p)). Explicit formulas are given for the eigenvalues and the energy of L(C(a))L(C(a)) where a=[a,a,…,a]a=[a,a,…,a]. Finally, a lower bound and an upper bound for the energy of L(C(p))L(C(p)) are derived.