The signless Laplacian spectral radii of modified graphs

In this paper, various modications of a connected graph G are regarded as perturbations of its signless Laplacian matrix. Several results concerning the resulting changes to the signless Laplacian spectral radius of G are obtained by solving intermediate eigenvalue problems of the second type

On the multiplicity of Laplacian eigenvalues of graphs

summary: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

A relation between multiplicity of nonzero eigenvalues and the matching number of graph

Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $\lambda$ of $A(G)$ is denoted by $m_\lambda(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$ at least $4$, then $m_\lambda(T)\leq \beta'(T)-1$ for any $\lambda\neq0$. Moreover, they characterized all trees with $m_\lambda(T)=\beta'(T)-1$, where $\beta'(G)$ is the induced matching number of $G$. In this paper, we intend to extend this result from trees to any connected graph. Contrary to the technique used in \cite{Wong}, we prove the following result mainly by employing algebraic methods: For any non-zero eigenvalue $\lambda$ of the connected graph $G$, $m_\lambda(G)\leq \beta'(G)+c(G)$, where $c(G)$ is the cyclomatic number of $G$, and the equality holds if and only if $G\cong C_3(a,a,a)$ or $G\cong C_5$, or a tree with the diameter is at most $3$. Furthermore, if $\beta'(G)\geq3$, we characterize all connected graphs with $m_\lambda(G)=\beta'(G)+c(G)-1$

Vulnerability Analysis of Soft Caving Tunnel Support System and Surrounding Rock Optimal Control Technology Research

The vulnerability assessment model, composed by 11 vulnerability factors, is established with the introduction of the concept of “vulnerability” into the assessment of tunnel support system. Analytic hierarchy process is utilized to divide these 11 factors into human attributes and natural attributes, and define the weight of these factors for the model. The “vulnerability” applied io the assessment of the tunnel support system model is reached. The vulnerability assessment model was used for evaluating and modifying the haulage tunnel #3207 of Bo-fang mine panel #2. The results decreased the vulnerability of the tunnel support system and demonstrated acceptable effects. Furthermore, the results show that the impact of human attributes on tunnel support systems is dramatic under the condition that natural attributes are permanent, and the “vulnerability” is exactly a notable factor to manifest the transformation during this process. The results also indicate that optimizing human attributes can attenuate vulnerability in tunnel support systems. As a result, enhancement of stability of tunnel support systems can be achieved

The algebraic connectivity of lollipop graphs

AbstractLet Cn,g be the lollipop graph obtained by appending a g-cycle Cg to a pendant vertex of a path on n-g vertices. In 2002, Fallat, Kirkland and Pati proved that for n⩾3g-12 and g⩾4, α(Cn,g)>α(Cn,g-1). In this paper, we prove that for g⩾4, α(Cn,g)>α(Cn,g-1) for all n, where α(Cn,g) is the algebraic connectivity of Cn,g

5,5′,5′′-Triphenyl-2,2′,2′′-[2,4,6-tri­methyl­benzene-1,3,5-triyltris(methyl­idene­sulfanedi­yl)]tris­(1,3,4-oxadiazole)

In the title compound, C36H30N6O3S3, the phenyl rings are twisted from the attached oxadiazole rings in the three arms by 1.5(2), 2.4 (2) and 25.7 (2)°. The crystal packing exhibits weak inter­molecular C—H⋯N inter­actions

5,5′-Diphenyl-2,2′-[butane-1,4-diylbis(sulfanedi­yl)]bis­(1,3,4-oxadiazole)

The complete mol­ecule of the title compound, C20H18N4O2S2, is generated by crystallographic inversion symmetry. The benzene ring is almost coplanar with the oxadiazole ring [dihedral angle = 7.2 (2)°]

On the Signless Laplacian Spectral Radius of Bicyclic Graphs with Perfect Matchings

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined