Estrada index of iterated line graphs

Tatjana Aleksić ; I. Gutman ; M. Petrović

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 32 (2007) no. 1.

Résumé

If $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of a graph $G$ , then the Estrada index of $G$ is $EE(G) = \sum\limits_{i=1}^n e^{\lambda_i}$ . If $L(G) = L^1(G)$ is the line graph of $G$ , then the iterated line graphs of $G$ are defined as $L^k(G) = L(L^{k-1}(G))$ for $k=2,3,\ldots$ . Let $G$ be a regular graph of order $n$ and degree $r$ . We show that $EE(L^k(G)) = a_k(r)\,EE(G) + n\,b_k(r)$ , where the multipliers $a_k(r)$ and $b_k(r)$ depend only on the parameters $r$ and $k$ . The main properties of $a_k(r)$ and $b_k(r)$ are established.