On Domination Number and Distance in Graphs

A vertex set $S$ of a graph $G$ is a \emph{dominating set} if each vertex of $G$ either belongs to $S$ or is adjacent to a vertex in $S$. The \emph{domination number} $\gamma(G)$ of $G$ is the minimum cardinality of $S$ as $S$ varies over all dominating sets of $G$. It is known that $\gamma(G) \ge \frac{1}{3}(diam(G)+1)$, where $diam(G)$ denotes the diameter of $G$. Define $C_r$ as the largest constant such that $\gamma(G) \ge C_r \sum_{1 \le i < j \le r}d(x_i, x_j)$ for any $r$ vertices of an arbitrary connected graph $G$; then $C_2=\frac{1}{3}$ in this view. The main result of this paper is that $C_r=\frac{1}{r(r-1)}$ for $r\geq 3$. It immediately follows that $\gamma(G)\geq \mu(G)=\frac{1}{n(n-1)}W(G)$, where $\mu(G)$ and $W(G)$ are respectively the average distance and the Wiener index of $G$ of order $n$. As an application of our main result, we prove a conjecture of DeLaVi\~{n}a et al.\;that $\gamma(G)\geq \frac{1}{2}(ecc_G(B)+1)$, where $ecc_G(B)$ denotes the eccentricity of the boundary of an arbitrary connected graph $G$.Comment: 5 pages, 2 figure

A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs

The \emph{metric dimension} $\dim(G)$ of a graph $G$ is the minimum number of vertices such that every vertex of $G$ is uniquely determined by its vector of distances to the chosen vertices. The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\!\setminus\!S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. We show that $\dim(T) \leq Z(T)$ for a tree $T$, and that $\dim(G) \le Z(G)+1$ if $G$ is a unicyclic graph, along the way, we characterize trees $T$ attaining $\dim(T)=Z(T)$. For a general graph $G$, we introduce the "cycle rank conjecture". We conclude with a proof of $\dim(T)-2 \leq \dim(T+e) \le \dim(T)+1$ for $e \in E(\overline{T})$.Comment: 15 pages, 14 figure

Domination in Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma(G)$ denote the domination number of $G$. It is readily seen that $\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.Comment: 18 pages, 8 figure

Identification of a Large Amount of Excess Fe in Superconducting Single-Layer FeSe/SrTiO3 Films

The single-layer FeSe films grown on SrTiO3 (STO) substrates have attracted much attention because of its record high superconducting critical temperature (Tc). It is usually believed that the composition of the epitaxially grown single-layer FeSe/STO films is stoichiometric, i.e., the ratio of Fe and Se is 1:1. Here we report the identification of a large amount of excess Fe in the superconducting single-layer FeSe/STO films. By depositing Se onto the superconducting single-layer FeSe/STO films, we find by in situ scanning tunneling microscopy (STM) the formation of the second-layer FeSe islands on the top of the first layer during the annealing process at a surprisingly low temperature ($\sim$150{\deg}C) which is much lower than the usual growth temperature ($\sim$490{\deg}C). This observation is used to detect excess Fe and estimate its quantity in the single-layer FeSe/STO films. The amount of excess Fe detected is at least 20% that is surprisingly high for the superconducting single-layer FeSe/STO films. The discovery of such a large amount of excess Fe should be taken into account in understanding the high-Tc superconductivity and points to a likely route to further enhance Tc in the superconducting single-layer FeSe/STO films