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
The algebraic connectivity of G is the second smallest eigenvalue of its Laplacian matrix. Let Un be the set of all unicyclic graphs of order n. In this paper, we will provide the ordering of unicyclic graphs in Un up to the last seven graphs according to their algebraic connectivities when n≥13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315-4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711]. 相似文献
In this paper we investigate the effect on the multiplicity of Laplacian eigenvalues of two disjoint connected graphs when
adding an edge between them. As an application of the result, the multiplicity of 1 as a Laplacian eigenvalue of trees is
also considered. 相似文献
The equilibrium electron transport of 1,4-diaminobenzene sandwiched between two Au electrodes is simulated by using a first principles analysis. The results show that equilibrium conductance increases with the molecule- electrode distance decreasing, and a platform occurs at the distance varying from 1.4 A to 1.9 A, implying the insensitiveness of 1,4-diaminobenzene equilibrium conductance to molecule-electrode distance. This is helpful to understand the improved reliability and reproducibility of conductance measurements using amines. 相似文献
CdSe:Eu nanocrystals were successfully synthesized and characterized by transmission electron microscopy, X-ray powder diffraction, and X-ray photoelectric spectroscopy. The CdSe:Eu nanocrystals showed enhanced green electrochemiluminescence (ECL) intensity when compared to pure CdSe nanocrystals. Further, the nanocrystals were used to design an ECL immunosensor for the detection of carcinoembryonic antigen (CEA) that has a linear response over the 1.0 fg·mL?1 to 100 ng·mL?1 CEA concentration range with a 0.4 fg·mL?1 detection limit. The assay was applied to the determination of CEA in human serum samples.
Graphical abstract Schematic of the assay: GCE-glassy-carbon electrode, Ab- Antibody, BSA- Bovine serum albumin, Ag- Antigen. CdSe:Eu nanocrystals were used to design an ECL immunosensor for the detection of carcinoembryonic antigen.
