Signless Laplacian spectral characterization of 4-rose graphs

A rose graph with p petals (or p-rose graph) is a graph obtained by taking p cycles with just a vertex in common. In this paper, we prove that all 4-rose graphs are determined by their signless Laplacian spectra.     

Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let C_a,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of C_a,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph C_a,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.     

Laplacian spectral characterization of clover graphs

A clover graph is obtained from a 3-rose graph by attaching a path to the vertex of degree six, where a 3-rose graph consists of three cycles with precisely one common vertex. In this paper, it is proved that all clover graphs are determined by their Laplacian spectra.     

Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.     

Signless Laplacian spectral radius and Hamiltonicity

We give tight conditions on the signless Laplacian spectral radius of a graph for the existence of Hamiltonian paths and cycles.     

Laplacian spectral characterization of 3-rose graphs

A 3-rose graph is a graph consisting of three cycles intersecting in a common vertex, J. Wang et al. showed all 3-rose graphs with at least one triangle are determined by their Laplacian spectra. In this paper, we complete the above Laplacian spectral characterization and prove that all 3-rose graphs are determined by their Laplacian spectra.     

The Laplacian spectral radius of some bipartite graphs

In this paper, we study the largest Laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs, respectively. Identifying the center of a star K_1,k and one vertex of degree n of K_m,n, we denote by the resulting graph. We show that the graph (1?k?n-4) is the unique graph with the largest Laplacian spectral radius among the bipartite graphs with n vertices and k cut edges, and (n?7) is the unique graph with the largest Laplacian spectral radius among all the bicyclic bipartite graphs.     

几类多圈图的拉普拉斯谱刻画

设图G是一个简单连通图. 如果任何一个与图G同拉普拉斯谱的图都与图G同构,则称图G是由其拉普拉斯谱确定的. 定义了双圈图\theta_{n}(p_1,p_2,\cdots,p_t) 和m 圈图H_n(m\cdot C_3;p_1,p_2,\cdots,p_t). 证明了双圈图\theta_{n}(p)和\theta_{n}(p,q),三圈图H_n(3\cdot C_3;p)和H_n(3\cdot C_3;p,q)分别是由它们的拉普拉斯谱确定的.     

Normalized Laplacian spectrum of some subdivision-coronas of two regular graphs

In this paper, we determine the full normalized Laplacian spectrum of the subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighbourhood corona and subdivision-edge neighbourhood corona of a connected regular graph with an arbitrary regular graph in terms of their normalized Laplacian eigenvalues. Moreover, applying these results, we find some non-regular normalized Laplacian cospectral graphs.     

On the spectral characterization of some unicyclic graphs

Let H(n;q,n₁,n₂) be a graph with n vertices containing a cycle C_q and two hanging paths P_n1 and P_n2 attached at the same vertex of the cycle. In this paper, we prove that except for the A-cospectral graphs H(12;6,1,5) and H(12;8,2,2), no two non-isomorphic graphs of the form H(n;q,n₁,n₂) are A-cospectral. It is proved that all graphs H(n;q,n₁,n₂) are determined by their L-spectra. And all graphs H(n;q,n₁,n₂) are proved to be determined by their Q-spectra, except for graphs with a being a positive even number and with b≥4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given.     

图的无符号拉普拉斯和距离无符号拉普拉斯特征值

Let G be a simple graph. We first show that ■, where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ(G), we determine the extremal graphs with the minimum ρQ(G) among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.     

The Laplacian spectral radius of graphs

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we improve Shi’s upper bound for the Laplacian spectral radius of irregular graphs and present some new bounds for the Laplacian spectral radius of some classes of graphs.     

On the signless Laplacian spectral characterization of the line graphs of T-shape trees

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic graph has the same signless Laplacian spectrum (simply G is DQS). Let T (a, b, c) denote the T-shape tree obtained by identifying the end vertices of three paths P _a+2, P _b+2 and P _c+2. We prove that its all line graphs L(T(a, b, c)) except L(T(t, t, 2t+1)) (t ? 1) are DQS, and determine the graphs which have the same signless Laplacian spectrum as L(T(t, t, 2t + 1)). Let µ₁(G) be the maximum signless Laplacian eigenvalue of the graph G. We give the limit of µ₁(L(T(a, b, c))), too.     

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.