The characteristic polynomial of a generalized join graph

A generalization of Fiedler's lemma and the spectra of H-join of graphs

“…, G k , where the subsets S i (G) ⊂ V (G i ) are arbitrary for 1 ≤ i ≤ k. Also, we deduce the characteristic polynomial of the generalized corona of graphs by visualizing corona as an H-join of graphs. Hence the results obtained (mostly for regular graphs) in [4,5,8,13,16,23,24], are generalized here for any graphs.…”

“…In [8,Theorem 2.3] another generalization of Fiedler's lemma, similar to Theorem 1, is given for the matrices with fixed row sum and the result is used to find the generalized characteristic polynomial of H-join of regular graphs. We observe that any such matrix has the all-one vector as an eigenvector.…”

“…The generalized characteristic polynomial of a graph G is introduced in [9], as the bivariate polynomial defined by φ G (λ, t) = det(λI − (A(G) − tD(G))) where A(G) and D(G) are the adjacency and the degree matrix associated to the graph G. As mentioned earlier, in [8,Theorem 3.1] the authors obtained a generalization of Fiedler's lemma, for the matrices with fixed row sum and as an application, they obtained the generalized characteristic polynomial of H-join of regular graphs. In the following theorem, we obtain the generalized characteristic polynomial of H-join of any graphs.…”

“…In [2], the author questioned the existence of non-regular graphs, which are cospectral with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian spectrum simultaneously. In [8,Theorem 3.7], the authors affirmatively answered the question by the construction of such graphs using the H-join of regular graphs. In the following theorem, we prove that those graphs are U -cospectral too.…”

“…In [14], the Laplacian spectra of H-join of any graphs is obtained. In [8] the characteristic polynomial of the matrix A(G) − tD(G) is obtained for H-join of regular graphs. In [23] the characteristic polynomial of the adjacency matrix of the lexicographic product of any graphs is obtained.…”