Affiliation:
1. Instituto de Matemática, Universidade Federal do Rio Grande do Sul , Porto Alegre , 91509-900 , Brazil
Abstract
Abstract
In this article, we study the Laplacian index of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle
C
k
{C}_{k}
and a path
P
n
{P}_{n}
. Using two different approaches, we show that their Laplacian index converges to
Δ
2
Δ
−
1
=
9
2
\frac{{\Delta }^{2}}{\Delta -1}=\frac{9}{2}
as
n
,
k
→
∞
n,k\to \infty
, where
Δ
=
3
\Delta =3
is the maximum degree of the graph. This limit is known as the Hoffman’s limit for the Laplacian matrix. The first technique is a linear time algorithm presented in [R. Braga, V. Rodrigues, and R. Silva, Locating eigenvalues of a symmetric matrix whose graph is unicyclic, Trends in Comput. Appl. Math. 22 (2021), no. 4, 659–674] that diagonalizes the matrix preserving its inertia. Here, we adapt this algorithm to the Laplacian index of a tadpole graph. The second method is to apply a formula presented in [V. Trevisan and E. R. Oliveira, Applications of rational difference equations to spectra graph theory, J. Difference Equ. Appl. 27 (2021), 1024–1051] for solving rational difference equations that appear when applying the diagonalization algorithm in some cases.