Analogously, we may define a diameter graph on the sphere S r of radius r. We think of the sphere being embedded in R, and the (unit) distance is induced from the ambient space. Diameter graphs are closely related to the famous Borsuk problem. In 1933 Borsuk [1] asked, whether any set of diameter 1 in R can be partitioned into (d+ 1) parts of strictly smaller diameter. The positive answer to th...

recently, e. m'{a}v{c}ajov'{a} and m. v{s}koviera proved that every bidirected eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. this result shows the validity of bouchet's nowhere zero conjecture for eulerian bidirected graphs. in this paper we prove the same theorem in a different terminology and with a short and simple proof. more precisely, we p...

Cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. Actually, the origin of this concept came back to Richard Brualdi's problems that are proposed in cite{braldi}: Let $G_n$ and $G'_n$ be two nonisomorphic simple graphs on $n$ vertices with spectra$$lambda_1 geq lambda_2 geq cdots geq lambda_n ;;;text{and};;; lambda'_1 geq lambda'_2...

A set D ⊆ V of vertices is said to be a (connected) distance k-dominating set of G if the distance between each vertex u ∈ V − D and D is at most k (and D induces a connected graph in G). The minimum cardinality of a (connected) distance k-dominating set in G is the (connected) distance k-domination number of G, denoted by γk(G) (γ c k (G), respectively). The set D is defined to be a total k-do...

it is necessary to generate the automorphism group of a chemical graph in computer-aidedstructure elucidation. an euclidean graph associated with a molecule is defined by a weightedgraph with adjacency matrix m = [dij], where for i≠j, dij is the euclidean distance between thenuclei i and j. in this matrix dii can be taken as zero if all the nuclei are equivalent. otherwise,one may introduce dif...

In 2005, Exoo posed the following question. Fix $\epsilon$ with $0\leq\epsilon<1$. Let $G_\epsilon$ be graph whose vertex set is Euclidean plane, where two vertices are adjacent iff distance between them lies in closed interval $[1-\epsilon,1+\epsilon]$. What chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ precisely classical ``chromatic plane'' problem. a 2018 p...