Some results on starlike and sunlike graphs
If \(G\) is a simple graph on vertices \(1,2,\dots,n\), the sunlike graph \(G(k_1,k_2,\dots,k_n)\) is obtained by identifying an end-vertex of a path \(P_{k_i}\), \(k_i\geq 1\), with vertex \(i\) of \(G\), \(i=1,2,\dots,n\). A formula for the characteristic polynomial of a sunlike graph is given and it is shown that, for \(n\geq 3\), whenever graphs \(K_n(k_1,k_2,\dots,k_n)\) and \(K_n(m_1,m_2,\dots,m_n)\) are cospectral, then they are also isomorphic. As a consequence, it is obtained that no two distinct starlike trees are Laplacian cospectral (a tree is starlike if it has exactly one vertex of degree greater than two).
scientific article; zbMATH DE number 1907720