Abstract Let G be a graph. Assume that to each vertex of set vertices $S\subseteq V(G)$ robot is assigned. At stage one can move neighbouring vertex. Then S mobile general position if there exists sequence moves the robots such all are visited while maintaining property at times. The number cardinality largest . We give bounds on and determine exact values for certain common classes graphs, inc...

-A vertex v V(G) is said to be a self vertex switching of G if G is isomorphic to G v , where G v is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. In [5], trees and forests are characterized, each with a self vertex switching. In [6], connected unicyclic graphs, each with a self vertex switching are characterized. In th...

a 2-emph{rainbow dominating function} (2rdf) on a graph $g=(v, e)$ is afunction $f$ from the vertex set $v$ to the set of all subsets of the set${1,2}$ such that for any vertex $vin v$ with $f(v)=emptyset$ thecondition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled. a 2rdf $f$ isindependent (i2rdf) if no two vertices assigned nonempty sets are adjacent.the emph{weight} of a 2rdf $f$ is the value $o...

Given a graph G, the burning number of G is smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ sequence G. It has been shown problem NP-complete, even trees with maximum degree three, or linear forests. A t-unicyclic unicycle in unique vertex greater than two $$ t + 2 . In this paper, we first present bounds graphs, and then use numbers ...

Let $g$ be a bounded symmetric measurable nonnegative function on $[0,1]^2$, and $\left\lVert g \right\rVert = \int_{[0,1]^2} g(x,y) dx dy$. For graph $G$ with vertices $\{v_1,v_2,\ldots,v_n\}$ edge set $E(G)$, we define
 \[ t(G,g) \; \int_{[0,1]^n} \prod_{\{v_i,v_j\} \in E(G)} g(x_i,x_j) \: dx_1 dx_2 \cdots dx_n .\]
 We conjecture that $t(G,g) \geq \left\lVert \right\rVert^{|E(G)|}$ ...

Let G = (V (G), E(G)) be a unicyclic simple undirected graph with largest vertex degree . Let Cr be the unique cycle of G. The graph G− E(Cr ) is a forest of r rooted trees T1,T2, . . .,Tr with root vertices v1, v2, . . ., vr , respectively. Let k(G) = max 1 i r {max{dist(vi , u) : u ∈ V (Ti )}} + 1, where dist(v, u) is the distance from v to u. Let μ1(G) and λ1(G) be the spectral radius of the...

A total vertex irregular k-labeling φ of a graph G is a labeling of the vertices and edges of G with labels from the set {1, 2, . . . , k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x. The minimum k for which the graph G...

Let G = (V,E) be a graph. The distance between two vertices u and v in a connected graph G is the length of the shortest (u−v) path in G. A set D ⊆ V (G) is a dominating set if every vertex of G is at distance at most 1 from an element of D. The domination number of G is the minimum cardinality of a dominating set of G. A set D ⊆ V (G) is a 2-distance dominating set if every vertex of G is at d...

The eccentric connectivity index ξ is a novel distance–based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as ξ(G) = ∑ v∈V (G) deg(v) · ε(v) , where deg(v) and ε(v) denote the vertex degree and eccentricity of v , respectively. We survey some mathematical properties of this index and furthermore support ...

Suppose the edges and the vertices of a simple graph G are assigned k-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there an...