Related papers: Average Nodal Count and the Nodal Count Condition …

We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$…

Let $G=\left( V\left( G\right) ,E\left( G\right) \right) $ be an $\left( n,m\right) $-graph and $X$ a nonempty proper subset of $V\left( G\right) $. Let $X^{c}=V\left( G\right) \backslash X$.\ The edge density of $X$ in $G$ is given by…

In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians, i.e., symmetric matrices with non-positive off-diagonal entries. In this paper, we establish nodal domain…

We prove that every metric graph which is a tree has an orthonormal sequence of Laplace-eigenfunctions of full support. This implies that the number of nodal domains $\nu_n$ of the $n$-th eigenfunction of the Laplacian with standard…

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional…

We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the…

We study the number of nodal components (connected components of the set of zeroes) of functions in the ensemble of arithmetic random waves, that is, random eigenfunctions of the Laplacian on the flat $d$-dimensional torus $\mathbb{T}^{d}$…

For a graph $G$, the generalized adjacency matrix $A_\alpha(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha) A(G)$ for $0\leq \alpha \leq 1$.…

A set $D$ of vertices of a graph $G$ is a dominating set if each vertex of $V(G)\setminus D$ is adjacent to some vertex of $D$. The domination number of $G$, $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. A graph $G$ is…

Let $G$ be a connected simple graph on $n$ vertices. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$ and $\rho_{n-1}(G)$ be the second least eigenvalue of $\mathcal{L}(G)$. Denote by $\nu(G)$ the independence number of $G$.…

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of…

We consider undirected simple finite graphs. The sets of vertices and edges of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$, we denote by $\delta(G)$ and $\eta(G)$ the least degree of a vertex of $G$ and the…

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le…

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if…

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 $T$…

Let $G$ be an $n$-vertex graph with Laplacian eigenvalues $0=\lambda_1(G)\le \lambda_2(G)\le\cdots\le \lambda_n(G)$. Motivated by the Alon-Boppana bound and the Ramanujan phenomenon for regular graphs, Spielman conjectured that, for every…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner…

The random intersection graph model $\mathcal G(n,m,p)$ is considered. Due to substantial edge dependencies, studying even fundamental statistics such as the subgraph count is significantly more challenging than in the classical binomial…

Let $G$ be a connected simple graph of order $n$. Let $\rho_1(G)\geq \rho_2(G)\geq \cdots \geq \rho_{n-1}(G)> \rho_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(\rho_i)$ the multiplicity…

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $\theta(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is…