Line graph eigenvalues and line energy of caterpillars

✍ Scribed by Oscar Rojo

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 504 KB
• Volume: 435
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2011.03.064

✦ Synopsis

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

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 ivertex 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.