Related papers: Graphs with span 1 and shortest optimal walks
In several recent papers, the maximal safety distance that two players can maintain while moving through a graph has been defined and studied using three different spans of the graph, each with different movement conditions. Mainly, vertex…
The strong vertex (edge) span of a given graph $G$ is the maximum distance that two players can maintain at all times while visiting all vertices (edges) of $G$ and moving either to an adjacent vertex or staying in the current position…
Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety…
The $d$-capacity of a graph $G$ is introduced as the maximum number of players that can simultaneously traverse $G$ such that each player visits all vertices while maintaining a distance of at least $d$ under various movement rules. We…
The work in this paper is motivated by I. Bani\v{c}'s and A. Taranenko's recent paper, where they introduced a new notion, the span of a graph. Their goal was to solve the problem of keeping a safety distance while two actors are moving…
It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common…
The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…
A drawing of a graph in the plane is called 1-planar if each edge is crossed at most once. A graph together with a 1-planar drawing is a 1-plane graph. A 1-plane graph $G$ with exactly $4|V (G)|-8$ edges is called optimal. The crossing…
Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ of $G$ is the maximum of the average distances…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…
The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of…
We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be…
We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a…
In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging…
Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum,…
We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the…
We give upper and lower bounds on the spectral radius of a graph in terms of the number of walks. We generalize a number of known results.
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…
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$,…
Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the…