On the Laplacian integral tricyclic graphs

Abstract

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.

Quadratic starlike trees

“…Then c k (x) = c S k (x) • q k (x) and q k (x)| f P k (x) ( for instance, q 6 (x) = f P 6 (x) and q 7 (x) = (x 4 − 4x 2 + 2), x(x 2 − 2)|c S 7 (x) and so on ). By Theorem 3.1 and Lemma 2.5 we have (10) where…”

“…Integral graphs are very rare and difficult to find, this reduces researchers to investigate integral graphs within restricted classes of graphs, such as starlike trees, balanced trees, some special classes of trees, trees with arbitrarily large diameters, graphs with few cycles and some Cayley graphs [2][3][4][5][6][7][8]. Also integral graphs are extended to various spectra such as Laplacian spectrum, Signless Laplacian spectrum [9,10]. One can refer to [3] for the summary of the researches of integral graphs.…”

On the Laplacian and signless Laplacian polynomials of graphs with semiregular automorphisms

“…One of the themes that has arisen in the literature on Laplacian (signless Laplacian) eigenvalues for graphs is that of Laplacian (signless Laplacian) integral graphs-i.e., those graphs whose all of their Laplacian (signless Laplacian) eigenvalues are integers. In [9,13,14,24,25,29], various families of Laplacian and signless Laplacian integral graphs were identified. Also, [16,17,28,30] provide constructions for certain classes of Laplacian and signless Laplacian integral graphs.…”

Q-integral unicyclic, bicyclic and tricyclic graphs

“…[16]) Let H be a tricyclic graph of order n with at least one pendant vertex. Then H is L-integral if and only if H ∈ {R, S, T, W }, where the corresponding graphs are shown in…”