Abstract
In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.
Highlights
All graphs considered in this paper are undirected and simple
We give the number of spanning trees, theKirchhoff index and the Kemeny’s constant of G1S . ( G2V ∪ G3E ), respectively
As an application of our main results (See Theorems 1–4), we focus the later and construct infinitely many pairs of M-cospectral mates (M = A, L, Q, L) since M-cospectral mates have the same
Summary
All graphs considered in this paper are undirected and simple. Let G = (V, E) be a graph with. Many graph operations such as the disjoint union, the corona [9], the edge corona [10,11], the neighborhood corona [12] and the subdivision vertex (edge) neighborhood corona [13] have been introduced, and their spectra are computed respectively. Indulal [14] introduced two new joins subdivision vertex-join G1 ∨ ̇ G2 and subdivision edge-join G1 Y G2 (for example, we depict P4 ∨ ̇ P3 and P4 Y P2 in Figure 1 if G1 = P4 and G2 = P3 , or P2 ), and their A-spectra are investigated when G1 and G2 are both regular graphs. We respectively determine the adjacency, the Laplacian and the signless Laplacian spectrum of G1S . “Which graph is determined by its spectrum?” [16] is a long-standing open problem in the theory of graph spectra.
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