Related papers: On a conjecture for the signless Laplacian eigenva…

Let $G$ be a simple connected graph on $n$ vertices and $m$ edges. In [Linear Algebra Appl. 435 (2011) 2570-2584], Lima et al. posed the following conjecture on the least eigenvalue $q_n(G)$ of the signless Laplacian of $G$: $\displaystyle…

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is…

Let G be a simple graph on $n$ vertices and $e(G)$ edges. Consider $Q(G) = D + A$ as the signless Laplacian of $G$, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of the vertices degree of $G$. Let $q_1(G)$ and $q_2(G)$ be…

Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq q_{n}(G) $ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G.$ In this paper we give a necessary and sufficient condition for the equality…

Let $G$ be a graph of order $n$ and let $q\left( G\right) $ be the largest eigenvalue of the signless Laplacian of $G$. Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an…

Let $G$ be a simple graph and $Q(G)$ be the signless Laplacian matrix of $G$. Let $S_\alpha(G)$ be the sum of the $\alpha$-th powers of the nonzero eigenvalues of $Q(G)$. We disprove two conjectures by You and Yang on the extremal values of…

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Very recently, Zhou, He and Shan proved that $K^+_{1,n-1}$ (the star graph with an additional edge) is the…

Let $q\left( G\right) $ be the $Q$-index (the largest eigenvalue of the signless Laplacian) of $G$. Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an independent set of order…

Let $G$ be a simple graph with the Laplacian matrix $L(G)$ and let $e(G)$ be the number of edges of $G$. A conjecture by Brouwer and a conjecture by Grone and Merris state that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at…

Let $G$ be a graph of order $n \geq 3$ with sequence degree given as $d_{1}(G) \geq ... \geq d_{n}(G)$ and let $\mu_1(G),..., \mu_n(G)$ and $q_1(G), ..., q_{n}(G)$ be the Laplacian and signless Laplacian eigenvalues of $G$ arranged in non…

We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let $\lambda_1(G)$ be the largest eigenvalue of the adjacency matrix of a graph $G$, and $\bar{G}$ be the complement of $G$.…

A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main…

Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be…

Brouwer's Conjecture states that, for any graph $G$, the sum of the $k$ largest (combinatorial) Laplacian eigenvalues of $G$ is at most $|E(G)| + \binom{k+1}{2}$, $1 \leq k \leq n$. We present several interrelated results establishing…

Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \#P3.6] showed that…

This paper gives tight upper bounds on the largest eigenvalue q(G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an…

This note presents a new spectral version of the graph Zarankiewicz problem: How large can be the maximum eigenvalue of the signless Laplacian of a graph of order $n$ that does not contain a specified complete bipartite subgraph. A…

Let $G$ be a simple graph with $n$ vertices and $m$ edges and let $k$ be a natural number such that $k\leq n.$ Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $m+{k+1 \choose 2}.$ In this paper we…

The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue…

Let $G=(V,E)$ be an $n$-vertex graph, $L(G)\in \mathbb{R}^{n\times n}$ its Laplacian matrix, and let $\lambda_1(L(G))\ge \lambda_2(L(G))\ge \cdots\ge \lambda_n(L(G))=0$ denote its eigenvalues. For $1\le k\le n$, let $\varepsilon_k(G)=…