On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Let G be a connected graph of order n and U a unicyclic graph with the same order. We firstly give a sharp bound for m_G(μ), the multiplicity of a Laplacian eigenvalue μ of G. As a straightforward result, m_U(1) ? n ? 2. We then provide two graph operations (i.e., grafting and shifting) on graph G for which the value of m_G(1) is nondecreasing. As applications, we get the distribution of m_U (1) for unicyclic graphs on n vertices. Moreover, for the two largest possible values of m_U(1) ∈ {n ? 5, n ? 3}, the corresponding graphs U are completely determined.

     

Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.     

The Laplacian spread of unicyclic graphs

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. In a recent work the trees with maximal Laplacian spread and with minimal Laplacian spread among all trees of fixed order are separately determined. In this work, we characterize the unique unicyclic graph with maximal Laplacian spread among all connected unicyclic graphs of fixed order.     

The minimum Laplacian spread of unicyclic graphs

The Laplacian spread of a graph [1] is defined as the difference between the largest eigenvalue and the second-smallest eigenvalue of the associated Laplacian matrix. In this paper, the minimum Laplacian spread of unicyclic graphs with given order is determined.     

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G)=\{v\mathrm{1},v\mathrm{2},\mathrm{...},vn\}. The transmission Tr(vi) (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n\times n diagonal matrix with its (i, i)-entry equal to {{\displaystyle Tr}}_G(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as L(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.     

A note on the signless Laplacian eigenvalues of graphs

In this paper, we consider the signless Laplacians of simple graphs and we give some eigenvalue inequalities. We first consider an interlacing theorem when a vertex is deleted. In particular, let G-v be a graph obtained from graph G by deleting its vertex v and κ_i(G) be the ith largest eigenvalue of the signless Laplacian of G, we show that κ_i+1(G)-1?κ_i(G-v)?κ_i(G). Next, we consider the third largest eigenvalue κ₃(G) and we give a lower bound in terms of the third largest degree d₃ of the graph G. In particular, we prove that . Furthermore, we show that in several situations the latter bound can be increased to d₃-1.     

On the sum of Laplacian eigenvalues of graphs

Let k be a natural number and let G be a graph with at least k vertices. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most , where e(G) is the number of edges of G. We prove this conjecture for k=2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G)+2k-1.     

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.     

On graphs with three distinct Laplacian eigenvalues

In this paper,an equivalent condition of a graph G with t(2≤t≤n)distinct Laplacian eigenvalues is established.By applying this condition to t=3,if G is regular(neces- sarily be strongly regular),an equivalent condition of G being Laplacian integral is given.Also for the case of t=3,if G is non-regular,it is found that G has diameter 2 and girth at most 5 if G is not a tree.Graph G is characterized in the case of its being triangle-free,bipartite and pentagon-free.In both cases,G is Laplacian integral.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let U _n,g be the lollipop graph obtained by appending a pendent vertex of a path on n ? g (n > g) vertices to a vertex of a cycle on g ? 3 vertices. We prove that the graph U _n,g uniquely minimizes the Laplacian spectral radius for n ? 2g ? 1 when g is even and for n ? 3g ? 1 when g is odd.     

On the signless Laplacian index of unicyclic graphs with fixed diameter

In the paper, we identify graphs with the maximal signless Laplacian spectral radius among all the unicyclicgraphs with n vertices of diameter d.