Abstract
A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to G. In some recent papers it is proved that the friendship graphs and starlike trees are DLS. If a friendship graph and a starlike tree are joined by merging their vertices of degree greater than two, then the resulting graph is called a path-friendship graph. In this paper, it is proved that the path-friendship graphs, a natural generalization of friendship graphs and starlike trees, are also DLS. Consequently, using these results we provide a solution for an open problem.
Highlights
Materialssome known results are given which are crucial throughout this paper
If a friendship graph and a starlike tree are joined by merging their vertices of degree greater than 2, the resulting graph is called a path-friendship graph
It is proved that all path-friendship graphs are DLS
Summary
Materialssome known results are given which are crucial throughout this paper. The lollipop graph of order n, denoted by Hn,p, is obtained by appending a cycle Cp to a pendant vertex of a path Pn−p. The aim of this paper is to prove that all path-friendship graphs are DLS. The following can be obtained from the Laplacian spectrum of a graph: i) the number of vertices, ii) the number of edges, iii) the number of spanning trees, iv) the number of components, v) the sum of the squares of the degrees of the vertices.
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