Related papers: The regular number of a graph

A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is…

The crossing number of a graph G is the minimum number of pairwise intersections of edges in a drawing of G. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…

We define the crossing number for an embedding of a graph G into R^3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the space crossing numbers of pseudo-random graphs.

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d(u,w)$ denote the length of a $u-w$ geodesic in $G$. For any $v\in V(G)$ and $e=xy\in E(G)$, let $d(e,v)=\min\{d(x,v),d(y,v)\}$. For distinct $e_1, e_2\in E(G)$, let…

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $\Delta(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph…

The boxicity of a graph $G$ is the minimum non-negative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space, where a box in Euclidean $k$-space is the Cartesian product of $k$…

Let $G$ be a simple graph. Consider all weightings of the vertices of $G$ with real numbers whose total sum is nonnegative. How many edges of $G$ have endpoints with a nonnegative sum? We consider the minimum number of such edges over all…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\Delta n)$, where $G$…

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. By appropriately introducing a weight to each edge of a graph, we determine, among other thing, the skewness of the generalized Petersen…

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…

An automorphism of a graph describes its structural symmetry and the concept of fixing number of a graph is used for breaking its symmetries (except the trivial one). In this paper, we evaluate automorphisms of the co-normal product graph…

Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of $G$ the set of colors appearing on the edges incident with $v$. The palette index of $G$ is defined as the minimum number of such distinct sets, taken over all…

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is…

Given a graph G, the domination number gamma(G) of G is the minimum order of a set S of vertices such that each vertex not in S is adjacent to some vertex in S. Equivalently, label the vertices from {0, 1} so that the sum over each closed…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

We consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed, undirected graph. We are motivated by the task of deriving lower bounds for consensus protocols and by the so-called "definitive…