Split non-threshold Laplacian integral graphs

The aim of this article is to answer a question posed by Merris in European Journal of Combinatorics, 24 (2003) pp. 413 ? 430, about the possibility of finding split non-threshold graphs that are Laplacian integral, i.e. graphs for which the eigenvalues of the corresponding Laplacian matrix are integers. Using Kronecker products, balanced incomplete block designs, and solutions to certain Diophantine equations, we show how to build infinite families of these graphs.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

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.     

Laplacian integral graphs in S(a, b)

Let G_n,m be the family of graphs with n vertices and m edges, when n and m are previously given. It is well-known that there is a subset of G_n,m constituted by graphs G such that the vertex connectivity, the edge connectivity, and the minimum degree are all equal. In this paper, S(a, b)-classes of connected (a, b)-linear graphs with n vertices and m edges are described, where m is given as a function of a,b∈N/2. Some of them have extremal graphs for which the equalities above are extended to algebraic connectivity. These graphs are Laplacian integral although they are not threshold graphs. However, we do build threshold graphs in S(a, b).     

不含三圈的k圈图的拟拉普拉斯和拉普拉斯谱半径

k圈图是边数等于顶点数加k-1的简单连通图.文中确定了不含三圈的k圈图的拟拉普拉斯谱半径的上界,并刻画了达到该上界的极图.此外,文中确定了拟拉普拉斯谱半径排在前五位的不含三圈的单圈图,排在前八位的不含三圈的双圈图.最后说明文中所得结论对不含三圈的k圈图的拉普拉斯谱半径也成立.     

On the Laplacian coefficients of bicyclic graphs

Let G be a graph of order n and let be the characteristic polynomial of its Laplacian matrix. Generalizing the approach in [D. Stevanovi?, A. Ili?, On the Laplacian coefficients of unicyclic graphs, Linear Algebra and its Applications 430 (2009) 2290-2300.] on graph transformations, we show that among all bicyclic graphs of order n, the kth coefficient c_k is smallest when the graph is B_n (obtained from C₄ by adding one edge connecting two non-adjacent vertices and adding n−4 pendent vertices attached to the vertex of degree 3).     

Spectral properties of Pascal graphs

We obtain spectral properties of the Pascal graphs by exploring its spectral graph invariants such as the algebraic connectivity, the first three largest Laplacian eigenvalues and the nullity. Some open problems pertaining to the Pascal graphs are given.     

On the Laplacian spectral radii of bicyclic graphs

A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n+1. Let B(n) be the set of all bicyclic graphs on n vertices. In this paper, we obtain the first four largest Laplacian spectral radii among all the graphs in the class B(n) (n≥7) together with the corresponding graphs.     

Indecomposable laplacian integral graphs

A graph that can be constructed from isolated vertices by the operations of union and complement is decomposable. Every decomposable graph is Laplacian integral. i.e., its Laplacian spectrum consists entirely of integers. An indecomposable graph is not decomposable. The main purpose of this note is to demonstrate the existence of infinitely many indecomposable Laplacian integral graphs.     

On the spectral radius of tricyclic graphs with a fixed diameter

A tricyclic graph is a connected graph with n vertices and n + 2 edges. In this article, we determine graphs with the largest spectral radius among the n-vertex tricyclic graphs with given diameter d.