Abstract
Let T be a tree with at least four vertices, none of which has degree 2, embedded in the plane. A Halin graph is a plane graph constructed by connecting the leaves of T into a cycle. Thus the cycle C forms the outer face of the Halin graph, with the tree inside it. Let G be a Halin graph with order n. Denote by mu(G) the Laplacian spectral radius of G. This paper determines all the Halin graphs with mu(G)geq n-4. Moreover, we obtain the graphs with the first three largest Laplacian spectral radius among all the Halin graphs on n vertices.
Highlights
In this paper, we consider simple and undirected connected graphs
Let G – v be the graph obtained from G by deleting the vertex v ∈ V (G)
We denote by Pn, Cn and Kn the path, cycle and complete graph on n vertices, respectively
Summary
Let G = G(V , E) be a simple graph with n vertices and m edges. Let G be a Halin graph on n vertices, μ(G) ≥ (G) + ≥ , the equality holds if and only if G ∼= W . ([ ]) Let G be a connected graph on n vertices with at least one edge. ([ ]) Let G be a graph and q(G) be the signless Laplacian spectral radius.
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