On simplices in diameter graphs in ℝ4

CENTERS OF TRIANGULATED GRAPHS V. D. Chepoi Let G = (X, U) be an ordinary graph with arbitrarily (not necessarily finitely) many vertexes, any two of which are joined by some finite chain. We endow G with a standard metric d(x, y) equal to the number of edges in the chain of shortest length joining vertexes x, y. The eccentricity e(z) of a vertex z is defined as {d(z, v): v e X}. The radius r(G) is the least eccentricity of the vertexes, and the diameter d(G) is the largest eccentricity. The center C(G) of G is the subgraph generated by the set of vertexes with minimal eccentricity. It is well known [I, 2] that any graph G, even if it is not connected, is the center of some graph G', i.e., G = C(G'). At the same time, if one confines attention to special classes of graphs, their centers may have rather specific features. Thus, a well-known re- sult of Jordan [3] states that the center of any finite tree consists of one vertex or two adjacent vertexes; that is to say, the only two possibilities are the graphs K l and K 2 (where K n denotes the complete subgraph on n vertexes). The centers of maximal outerplanar graphs and 2-trees were described in [4, 5]. In this paper we characterize the centers of triangu- lated graphs. In this connection, we note that metric properties, including in particular properties of the centers, of triangulated graphs have been studied by various authors [6-9]. Our Theorems 1 and 2 were proved in [8], but they are established here in a more general form and the proofs are simpler. Recall [i0] that a graph G is said to be triangulated if, in any simple cycle F of length more than 3, there are two vertexes not adjacent in r but joined by an edge in G. Recall moreover that a clique of a graph G is any maximal complete subgraph (with re- spect to inclusion). The density of a graph G is the cardinality of the largest clique in G (if such a clique exists). Throughout this paper, the term triangulated graph will mean a triangulated graph of finite diameter without infinite complete subgraphs. Let CT denote the family of graphs which are centers of triangulated graphs. We shall need a number of additional concepts and definitions. Given sets M c X in a graph G and any number r ~ 0, we put where

Let x ∈ ℤn and AI(x) = {r ∈ ℤn : rx ∈ I}. The annihilator ideal graph of ℤn, denoted by AGI (ℤn) is the undirected graph with vertex set V(AGI(ℤn))={ x∈ℤn\I:xy∈I,y∉I }, and two distinct vertices x and y are adjacent if and only if AI(xy) ≠ AI(x)∪AI(y). In this paper, we study the diameter of annihilator ideal graph of ℤn, where ℤn is a commutative ring of integers modulo n.

On isometric subgraphs of infinite bridged graphs and geodesic convexity

We prove that, for fixed k ≥ 3, the following classes of labeled n-vertex graphs are asymptotically equicardinal: graphs of diameter k, connected graphs of diameter at least k, and (not necessarily connected) graphs with a shortest path of length at least k. An asymptotically exact approximation of the number of such n-vertex graphs is obtained, and an explicit error estimate in the approximation is found. Thus, the estimates are improved for the asymptotic approximation of the number of n-vertex graphs of fixed diameter k earlier obtained by Füredi and Kim. It is shown that almost all graphs of diameter k have a unique pair of diametrical vertices but almost all graphs of diameter 2 have more than one pair of such vertices.

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.

The center (periphery) of a graph is the set of vertices with minimum (maximum) eccentricity. In this paper, the structure of centers and peripheries of some classes of composite graphs are determined. The relations between eccentricity, radius and diameter of such composite graphs are also investigated. As an application we determine the center and periphery of some chemical graphs such as nanotorus and nanotubes covered by C4.

Cayley graphs have been well-studied as a model for interconnection networks due to their low diameter, optimal fault tolerance, and algorithmic efficiency, among other properties. A problem of practical and theoretical interest is to determine or estimate the diameter of Cayley graphs. Let Γ be a Cayley graph on n! vertices generated by a transposition tree on vertex set {1,2,…,n}. In an oft-cited paper [1], it was shown that the diameter of Γ is bounded as: $$\mathop{\mathrm{diam}}\nolimits (\Gamma)~ \le ~\max_{\pi \in S_n} \left\{ c(\pi)-n+\sum_{i=1}^n \mathop{\mathrm{dist}}\nolimits _T(i,\pi(i)) \right\},$$ where the maximization is over all permutations π in the symmetric group, c(π) denotes the number of cycles in π, and \(\mathop{\mathrm{dist}}\nolimits _T\) is the distance function in T. It is of interest to determine how far away this upper bound can be from the true diameter value in the worst case and for which families of graphs this bound should be utilized or not utilized. In this work, we investigate the worst case performance of this upper bound. We show that for every n, there exists a transposition tree on n vertices such that the maximum possible difference Δ n between the upper bound and the true diameter value is at least n − 4. The lower bound we provide for Δ n is seen to be best possible, and an open problem is to determine an upper bound for Δ n .

Application of coding theory to interconnection networks

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant $$d_\infty $$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on $${{\mathbf {R}}}^n$$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $$1/(n+1)$$ to each vertex of a (unit-diameter) regular simplex.

Let G = (V(G),E(G)) denote a graph whose set of vertices and set of edges are V(G)and E(G), respectively. For any v ∈ V(G), we denote the neighbors of v as N(v) .B yn(G), we denote the number of vertices of G. All graphs considered here are both finite and simple. We denote, respectively, by Sn, Pn, and Cn the star, path, and cycle with n vertices. Let (G1,v1) and (G2,v2) be two graphs rooted at v1 and v2, respectively, then G = (G1,v1) �� (G1,v2) denote the graph obtained by identifying v1 with v2 as one common vertex. Let Un denote the set of all unicycle graphs of order n .B yU(n,k) we denote the set of unicycle graphs in which the length of its cycle is k. For any graph G in U(n,k), we denote the unique cycle of length k in G as Ck. Other notations and terminology not defined here will conform to those in [1]. For any given graph G, its Merrifield‐Simmons index, simply denoted as f(G), is defined to be the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., in graph-theoretical terminology, the number of independent-vertex subsets of G, including the empty set. We take the cycle C4 =

A family of circuits of a graphGis said to beindependentif no two of the circuits have a common vertex; it is callededge-independentif no two of them have an edge in common. A set of vertices will be called arepresentingset for the circuits (for the sake of brevity we shall call it a representing set), if every circuit ofGpasses through at least one vertex of the representing set. Denote byI(G) =kthe maximum number of circuits in an independent family and byR(G) the minimum number of vertices of a representing set. Dirac and Gallai asked whether there is any relation betweenI(G) andR(G) (triviallyR(G) ≥I(G)).

We introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex set of G into cliques and a tree T having P as a vertex set such that for distinct X,Y∈P, (1) the edges between X and Y in G form a complete bipartite subgraph whose parts are some subsets of X and Y, if X and Y are adjacent in T, and (2) there are no edges between X and Y in G otherwise. A split graph with vertex partition (C,I) where C is a clique and I is an independent set is a well-partitioned chordal graph as witnessed by a star T having C as the center and each vertex in I as a leaf, viewed as a clique of size 1. We characterize well-partitioned chordal graphs by forbidden induced subgraphs, and give a polynomial-time algorithm that given a graph, either finds an obstruction, or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal.We observe that there are problems, for instance Densestk-Subgraph and b-Coloring, that are polynomial-time solvable on split graphs but become NP -hard on well-partitioned chordal graphs. On the other hand, we show that the Geodetic Set problem, known to be NP -hard on chordal graphs, can be solved in polynomial time on well-partitioned chordal graphs.We also answer two combinatorial questions on well-partitioned chordal graphs that are open on chordal graphs, namely that each well-partitioned chordal graph admits a polynomial-time constructible tree 3-spanner, and that each (2-connected) well-partitioned chordal graph has a vertex that intersects all its longest paths (cycles).

The nilpotent graph is a graph’s structure motivated by the characteristics of some elements of a ring. If G is a graph, G(N) is a nilpotent graph in which the set of all non-nilpotent elements of a ring taken as a vertex set. For each x, y in R\ il(R), x and y are said to be connected or adjacent to an edge, if the sum of the vertices x and y is equal to the nilpotent element in the ring. This research aims to explain how to construct a nilpotent graph G(N) from a commutative ring. The results of this research are the characteristic of the nilpotent graph of a ring Zn for n prime numbers, a nilpotent graph for the ring Z 2 k, and ring Zn with n = 6m where 1 ≤ m < 5 can create 3 nilpotent subgraphs, they are a complete subgraph (km ) and 2 regular subgraphs in which the degree of vertices is equal to m.

Given an undirected graph G=(V,E) with vertex set V={1,?,n} and edge set E?V×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.

Let G be a connected, undirected graph. Distance two labeling or a L(2,1)-labeling of a graph G is an assignment f from the vertex set V (G) to the set of non- negative integers such that |f(x)− f(y)| is greater than or equal to 2 if x and y are adjacent and |f(x) − f(y)| is greater than or equal to 1 if x and y are at distance 2, for all x and y in V (G). The L(2,1)-labeling number (Lamda)(G) of G is the smallest number k such that G has an L(2,1)-labeling f with max {f(v) : v 2 V (G) }is equal to k. In this paper, we present algorithms to get L(2,1)-labeling of cycle dominating graphs like Diamond graphs, nC4 with a common vertex and Books Bn and hence we find the (lamda)- number of these graphs. Keywords: Channel assignment; transmitters; L(2,1)-labeling; Books; Distance two labeling 2010 Mathematics Subject Classification: 05C78.