This paper introduces the notion of informative labeling schemes for arbitrary graphs. Let f (W) be a function on subsets of vertices W . An f labeling scheme labels the vertices of a weighted graph G in such a way that f (W) can be inferred (or at least approximated) efficiently for any vertex subset W of G by merely inspecting the labels of the vertices of W, without having to use any additio...

A Steiner star for a set P of n points in R connects an arbitrary center point to all points of P , while a star connects a point p ∈ P to the remaining n−1 points of P . All connections are realized by straight line segments. Fekete and Meijer showed that the minimum star is at most √ 2 times longer than the minimum Steiner star for any finite point configuration in R. The maximum ratio betwee...

Steiner and Schwarz symmetrizations, their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, outer rotational are investigated. The focus is on convergence of successive symmetrals with respect to a sequence $i$-dimensional subspaces $\mathbb{R}^n$. Such called universal for family sets if any set in converge ball center at origin. New sequences main all vali...

There are a number of positional games known on the infinite chessboard. One of the most studied is the 5-in-a-row, whose rules are almost identical to the ancient Japanese Go-Moku. Along this line Harary asked if a player can achieve a translated copy of a given polymino P when the two players alternately take the squares of the board. Here we pose his question for general subsets of the board...

In 1994, Dobrynin and Kochetova introduced the concept of degree distance DD(Γ) a connected graph Γ. Let dΓ(S) be Steiner k-distance S⊆V(Γ). The Wiener k-index or k-center indexSWk(Γ) Γ is defined by SWk(Γ)=∑|S|=kS⊆V(Γ)dΓ(S). distanceSDDk(Γ) SDDk(Γ)=∑|S|=kS⊆V(Γ)∑v∈SdegΓ(v)dΓ(S), where degΓ(v) vertex v in this paper, we consider Nordhaus–Gaddum-type results for SWk(Γ) SDDk(Γ). Upper bounds on SW...

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) the diagonal degrees and RD(G) Harary matrix. Clearly, RDL(G) real symmetric matrix, we denote its eigenvalues λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). largest eigenvalue λ1(RDL(G)) RDL(G), denoted by λ(G), called spectral radius. In this paper, obtain several upper bounds for sum k in terms...

the stability of a communication network composed of processing nodes and communication links is of prime importance to network designers. as the network begins losing links or nodes, eventually there is a loss in its effectiveness. thus, communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible...