Related papers: Nordhaus--Gaddum type inequalities for Laplacian a…

Let $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$. The vertex connectivity of $G$, written $\kappa(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $\mu_2(G) \le…

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not…

Let $G$ be a simple undirected graph. For $\alpha \in [0,1]$, let \begin{equation*} A_{\alpha}\left( G\right) =\alpha D\left( G\right) +(1-\alpha)A\left( G\right) , \end{equation*} where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is…

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$ and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as…

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a…

If $G$ is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs $G_{1}$ and $G_{2}$ is a graph $G=G_{1}\odot G_{2}$ with $V(G)=V(G_{1})\cup…

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all outerplanar graphs on $n$…

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a…

{Signless Laplacian determinations of some graphs with independent edges}% {Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the…

Let $G=(V,E)$ be a graph on $n$ vertices, and let $\lambda_1(L(G))\ge \cdots\ge \lambda_{n-1}(L(G))\ge \lambda_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k…

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain…

Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless…

Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of…

A $k$-decomposition $(G_1,\dots,G_k)$ of a graph $G$ is a partition of its edge set into $k$ spanning subgraphs $G_1,\dots,G_k$. The classical theorem of Nordhaus and Gaddum bounds $\chi(G_1) + \chi(G_2)$ and $\chi(G_1) \chi(G_2)$ over all…

Let $G$ be a simple graph. The signless Laplacian spread of $G$ is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian…

Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i \varphi_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations…

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…

A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let $C_l$, $P_l$, $K_l$ and $K_{s,l-s}$ be the cycle, the path, the complete graph and the…

The spread of a graph $G$ is the difference $\lambda_1 - \lambda_n$ between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on $n$ vertices with maximum spread…

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…