Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \geq 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.
n this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with n vertices and clique number ω(2 ≤ ω ≤ n) are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved., Kinkar Ch. Das, Muhuo Liu., and Obsahuje seznam literatury
Let $W_{n}=K_{1}\vee C_{n-1}$ be the wheel graph on $n$ vertices, and let $S(n,c,k)$ be the graph on $n$ vertices obtained by attaching $n-2c-2k-1$ pendant edges together with $k$ hanging paths of length two at vertex $v_{0}$, where $v_{0}$ is the unique common vertex of $c$ triangles. In this paper we show that $S(n,c,k)$ ($c\geq 1$, $k\geq 1$) and $W_{n}$ are determined by their signless Laplacian spectra, respectively. Moreover, we also prove that $S(n,c,k)$ and its complement graph are determined by their Laplacian spectra, respectively, for $c\geq 0$ and $k\geq 1$.
In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with $n$ vertices and $k$ pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with $n$ vertices and $k$ pendant vertices, respectively.