Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

Abstract

If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graphIn this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type G 1 uv G 2 . We also provide a condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. Moreover, we characterize the broken sun graphs and the one-edge connection of two broken sun graphs by their Laplacian eigenvalue 2.

The multiplicity of the Laplacian eigenvalue $2$ in some bicyclic graphs

“…In [3] authors have considered a necessary and sufficient condition in the bicyclic graph G = G 1 ⊙ uv G 2 for having the Laplacian eigenvalue 2, where G 1 and G 2 are unicyclic graphs and have 2…”

“…. , x n1 ) t , such that [3,Theorem 4], so x(u) = 0. By contrary, if G has 2 among its Laplacian eigenvalues, then we can assume that X = (x 1 , .…”

“…and the result follows. ✷In[3], we have shown the upper bound for m G (λ), where λ > 1.Lemma 4. [3, Lemma 2] Let G be a bicyclic graph and λ > 1 be an integral eigenvalue of L(G).…”