For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the metric representation of v with respect to W, where d(x, y) is the distance between vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we characterize all graphs of order n with metric dimension n - 3.
A Lie algebra $L$ is called 2-step nilpotent if $L$ is not abelian and $[L, L]$ lies in the center of $L$. 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center., Ren Bin, Zhu Lin Sheng., and Obsahuje bibliografii
For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace $ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1)$, $d(v, w_2),\dots ,d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $\dim (G)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph $<W>$ induced by $W$ is a nontrivial connected subgraph of $G$. The minimum cardinality of a connected resolving set in a graph $G$ is its connected resolving number $\mathop {\mathrm cr}(G)$. Thus $1 \le \dim (G) \le \mathop {\mathrm cr}(G) \le n-1$ for every connected graph $G$ of order $n \ge 3$. The connected resolving numbers of some well-known graphs are determined. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $\mathop {\mathrm cr}(G) = n-1$ if and only if $G = K_n$ or $G = K_{1, n-1}$. It is also shown that for positive integers $a$, $b$ with $a \le b$, there exists a connected graph $G$ with $\dim (G) = a$ and $\mathop {\mathrm cr}(G) = b$ if and only if $(a, b) \notin \lbrace
(1, k)\: k = 1\hspace{5.0pt}\text{or}\hspace{5.0pt}k \ge 3\rbrace $. Several other realization results are present. The connected resolving numbers of the Cartesian products $G \times K_2$ for connected graphs $G$ are studied.
Some observations concerning McShane type integrals are collected. In particular, a simple construction of continuous major/minor functions for a McShane integrand in Rn is given.
The basis number of a graph $G$ was defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. He proved that for $m,n\ge 5$, the basis number $b(K_{m,n})$ of the complete bipartite graph $K_{m,n}$ is equal to 4 except for $K_{6,10}$, $K_{5,n}$ and $K_{6,n}$ with $n=5,6,7,8$. We determine the basis number of some particular non-planar graphs such as $K_{5,n}$ and $K_{6,n}$, $n=5,6,7,8$, and $r$-cages for $r=5,6,7,8$, and the Robertson graph.
For an ordered set W = {w1, w2,...,wk} of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v|W)=(d(v,w1), d(v, w2),...,d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations. A resolving set of minimum cardinality is a basis for G and the number of vertices in a basis is its (metric) dimension dim(G). For a basis W of G, a subset S of W is called a forcing subset of W if W is the unique basis containing S. The forcing number fG(W, dim) of W in G is the minimum cardinality of a forcing subset for W, while the forcing dimension f(G, dim) of G is the smallest forcing number among all bases of G. The forcing dimensions of some well-known graphs are determined. It is shown that for all integers a, b with 0 ≤ a ≤ b and b ≥1, there exists a nontrivial connected graph G with f(G) = a and dim(G) = b if and only if {a, b} ≠ {0, 1}.