Graphs with three mutually pseudo-similar vertices

Vertices u and v of a graph X are pseudo-similar if X ? u ? X ? v but no automorphism of X maps u to v. We describe a group-theoretic method for constructing graphs with a set of three mutually pseudo-similar vertices. The method is used to construct several examples of such graphs. An algorithm for extending, in a natural way, certain graphs with three mutually pseudo-similar vertices to a graph in which the three vertices are similar is given. The algorithm suggests that no simple characterization of graphs with a set of three mutually pseudo-similar vertices can exist.     

On graphs with cut vertices and cut edges

Let G（n,k,t） be a set of graphs with n vertices,k cut edges and t cut vertices.In this paper,we classify these graphs in G（n,k,t） according to cut vertices,and characterize the extremal graphs with the largest spectral radius in G（n,k,t）.     

Constructing graphs with pairs of pseudo-similar vertices

Vertices u and v in the graph G are said to be pseudo-similar if G ? u ? G ? v but no automorphism of G maps u onto v. It is shown that a known procedure for constructing finite graphs with pairs of pseudo-similar vertices actually produces all such graphs. An additional procedure for constructing infinite graphs with pseudo-similar vertices is introduced and it is shown that all such graphs can be obtained by using either this or the first-mentioned method. The corresponding result for pseudo-similar edges is given.     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

On the nullity of graphs with pendent vertices

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60-67] characterized the extremal graphs attaining the upper bound n-2 and the second upper bound n-3. In this paper, as the continuance of it, we determine the extremal graphs with pendent vertices achieving the third upper bound n-4 and fourth upper bound n-5. We then proceed recursively to construct all graphs with pendent vertices which satisfy η(G)>0. Our results provide a unified approach to determine n-vertex unicyclic (respectively, bicyclic and tricyclic) graphs which achieve the maximal and second maximal nullity and characterize n-vertex extremal trees attaining the second and third maximal nullity. As a consequence we, respectively, determine the nullity sets of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs on n vertices.     

On the spectral radius of bicyclic graphs with n vertices and diameter d

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d. As an application, we give first three graphs among all bicyclic graphs on n vertices, ordered according to their spectral radii in decreasing order.     

Bounds on the eigenvalues of graphs with cut vertices or edges

In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices.     

On the signless Laplacian spectral radius of graphs with cut vertices

In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph G_n,k, where G_n,k is obtained from the complete graph K_n-k by attaching paths of almost equal lengths to all vertices of K_n-k. We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see [A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233-240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius.     

Some results on Laplacian spectral radius of graphs with cut vertices

In this paper, we give some results on Laplacian spectral radius of graphs with cut vertices, and as their applications, we also determine the unique graph with the largest Laplacian spectral radius among all unicyclic graphs with n vertices and diameter d, 3?d?n−3.     

Note on unicyclic graphs with given number of pendent vertices and minimal energy

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let G(n,p) denote the set of unicyclic graphs with n vertices and p pendent vertices. In [H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Linear Algebra Appl. 426 (2007) 478-489], Hua and Wang discussed the graphs that have minimal energy in G(n,p), and determined the minimal-energy graphs among almost all different cases of n and p. In their work, certain case of the values was left as an open problem in which the minimal-energy species have to be chosen in two candidate graphs, but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials. This paper aims at solving the problem completely, by using the well-known Coulson integral formula.     

The crossing numbers of join of the special graph on six vertices with path and cycle

There are only few results concerning crossing numbers of join of some graphs. In the paper, for the special graph H on six vertices we give the crossing numbers of its join with n isolated vertices as well as with the path P_n on n vertices and with the cycle C_n.     

Coverings of the complete (di-)graph with n vertices by complete bipartite (di-)graphs with n vertices I

Let n be an integer, n ? 2. A set $\text{M}$_n of complete bipartite (di-)graphs with n vertices is called a critical covering of the complete (di-)graph with n vertices if and only if the complete (di-)graph is covered by the (di-)graphs of $\text{M}$_n, but of no proper subset of $\text{M}$_n. All possible cardinalities of critical coverings $\text{M}$_n are determined for all integers n and for undirected as well as directed graphs.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

On the signless Laplacian index of cacti with a given number of pendant vertices

A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this article, we determine graphs with the largest signless Laplacian index among all the cacti with n vertices and k pendant vertices. As a consequence, we determine the graph with the largest signless Laplacian index among all the cacti with n vertices; we also characterize the n-vertex cacti with a perfect matching having the largest signless Laplacian index.     

Partition the vertices of a graph into one independent set and one acyclic set

For a given graph G, if the vertices of G can be partitioned into an independent set and an acyclic set, then we call G a near-bipartite graph. This paper studies the recognition of near-bipartite graphs. We give simple characterizations for those near-bipartite graphs having maximum degree at most 3 and those having diameter 2. We also show that the recognition of near-bipartite graphs is NP-complete even for graphs where the maximum degree is 4 or where the diameter is 4.     

On the number of noncritical vertices in strongly connected digraphs

By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.     

Pseudosimilar vertices in a graph

Dissimilar vertices whose removal leaves isomorphic subgraphs are called pseudosimilar. We construct infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex. Potential impact on the Reconstruction Conjecture is considered. We also construct, for each n, graphs containing a subset of n vertices which are mutually pseudosimilar. the analogous problem for mutually pseudosimilar edges is introduced.