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1.
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. We obtain some lower bounds for the nullity of graphs and we then find the nullity of bipartite graphs with no cycle of length a multiple of 4 as a subgraph. Among bipartite graphs on n vertices, the star has the greatest nullity (equal to n − 2). We generalize this by showing that among bipartite graphs with n vertices, e edges and maximum degree Δ which do not have any cycle of length a multiple of 4 as a subgraph, the greatest nullity is . G. R. Omidi: This research was in part supported by a grant from IPM (No.87050016).  相似文献   

2.
The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G)?n-2 if G is a simple graph on n vertices and G is not isomorphic to nK1. The extremal graphs attaining the upper bound n-2 and the second upper bound n-3 have been obtained. In this paper, the graphs with nullity n-4 are characterized. Furthermore the tricyclic graphs with maximum nullity are discussed.  相似文献   

3.
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.  相似文献   

4.
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we obtain the nullity set of bicyclic graphs of order n, and determine the bicyclic graphs with maximum nullity.  相似文献   

5.
A short proof is given of the impossibility of decomposing the complete graph on n vertices into n-2 or fewer complete bipartite graphs.  相似文献   

6.
For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n×n matrices A whose (i,j)th entry (for ij) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is introduced. Maximum generic nullity provides insight into the structure of the null-space of a matrix realizing maximum nullity of a graph. It is shown that maximum generic nullity is bounded above by edge connectivity and below by vertex connectivity. Results on random graphs are used to show that as n goes to infinity almost all graphs have equal maximum generic nullity, vertex connectivity, edge connectivity, and minimum degree.  相似文献   

7.
Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph GΩ is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.  相似文献   

8.
A graph G is diameter 2-critical if its diameter is 2, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an important association with total domination to prove the conjecture for the graphs whose complements are claw-free.  相似文献   

9.
A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. In this paper, we determine the maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartite graphs. © 1993 John Wiley & Sons, Inc.  相似文献   

10.
A graph G is diameter 2-critical if its diameter is two, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an association with total domination to prove the conjecture for the graphs whose complements have diameter three.  相似文献   

11.
The notion of partitional graphs, a subclass of sequential graphs, is introduced, and the cartesian product of a partitional graph and K 2 is shown to be partitional. Every sequential graph is harmonious and felicitous. The partitional property of some bipartite graphs including the n-dimensional cube Q n is studied, and thus this paper extends what was known about the sequentialness, harmoniousness and felicitousness of such graphs.  相似文献   

12.
A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+F is a chain graph (i.e., a 2K 2-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n 2) recognition algorithm.  相似文献   

13.
We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c > 0 such that for every family F of n x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turán-type results on incomparability graphs of partially ordered sets.  相似文献   

14.
A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if the deletion of each minimum vertex cut creates exactly two components, one of which is an isolated vertex. In this note, we proved that a vertex transitive bipartite graph is not super-connected if and only if it is isomorphic to the lexicographic product of a cycle Cn(n ≥ 6) by a null graph Nm. We also characterized non-hyper-connected vertex transitive bipartite graphs.  相似文献   

15.
A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.  相似文献   

16.
Maximal Energy Bipartite Graphs   总被引:1,自引:0,他引:1  
 Given a graph G, its energy E(G) is defined to be the sum of the absolute values of the eigenvalues of G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. Here we show that if G is a bipartite graph with n vertices, then
must hold, characterize those bipartite graphs for which this bound is sharp, and provide an infinite family of maximal energy bipartite graphs. Received: December 1, 2000 Final version received: August 28, 2001 RID="*" ID="*" The author thanks the Swedish Natural Science Research Council (NFR) – grant M12342-300 – for its support. Acknowledgments. The authors would like to thank Ivan Gutman for encouraging them to write this paper, and for helpful discussions on this topic. They also would like to thank Edwin van Dam for his reference concerning connected bipartite regular graphs with four eigenvalues.  相似文献   

17.
Let G be a graph with n vertices and ν(G) be the matching number of G. Let η(G) denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G)=n-2ν(G). Tan and Liu [X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Alg. Appl. 408 (2005) 212-220] proved that the nullity set of all unicyclic graphs with n vertices is {0,1,…,n-4} and characterized the unicyclic graphs with η(G)=n-4. In this paper, we characterize the unicyclic graphs with η(G)=n-5, and we prove that if G is a unicyclic graph, then η(G) equals , or n-2ν(G)+2. We also give a characterization of these three types of graphs. Furthermore, we determine the unicyclic graphs G with η(G)=0, which answers affirmatively an open problem by Tan and Liu.  相似文献   

18.
The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.  相似文献   

19.
令$\eta(\Gamma)$和$c(\Gamma)$是符号图$\Gamma$的零度和基本圈数. 一个符号圈拼接图是指每个块都是圈的连通符号图. 本文证明了对任意符号拼接图$\eta(\Gamma)\le c(\Gamma)+1$成立, 并且刻画了等号成立的极图, 推广了王登银等人(2022)在简单圈拼接图上的结果. 此外, 我们证明了任意的符号拼接图$\eta(\Gamma)\neq c(\Gamma)$, 给出了满足$\eta(\Gamma)=c(\Gamma)-1$的符号拼接图的一些性质并刻画处$\eta(\Gamma)=c(\Gamma)-1$的二部符号拼接图.  相似文献   

20.
We show new lower and upper bounds on the maximum number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105n/10 ≈ 1.5926n; such subgraphs show an upper bound of O(12n/4) = O(1.8613n) and give an algorithm that finds all maximal induced bipartite subgraphs in time within a polynomial factor of this bound. This algorithm is used in the construction of algorithms for checking k‐colorability of a graph. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 127–132, 2005  相似文献   

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