Related papers: Maxima of the Q-index: degenerate graphs

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…

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$…

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…

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix…

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…

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…

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is…

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 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 simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \ldots, n.$ F. Ashraf et al.…

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…

The $Q$-index of graph $G$ is the largest eigenvalue of the signless Laplacian matrix of $G$. Wang [Discrete Appl. Math. 356(2024)] proved the sharp upper bounds on the $Q$-index of leaf-free graphs with given size and characterized the…

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the…

For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex…

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha…

Let $\lambda(G)$ be the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed from…

Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…

If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote…

Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $\Delta$. Let $\lambda_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the…

A well known upper bound for the independence number $\alpha(G)$ of a graph $G$, due to Cvetkovi\'{c}, is that \begin{equation*} \alpha(G) \le n^0 + \min\{n^+ , n^-\} \end{equation*} where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove…