Showing papers on "Path graph published in 1982"

Largest random component of a k -cube

Abstract: LetC k denote the graph with vertices (ɛ 1, ...,ɛ k ),ɛ i =0,1 and vertices adjacent if they differ in exactly one coordinate. We callC k thek-cube. LetG=G k, p denote the random subgraph ofC k defined by letting $$Prob(\{ i,j\} \in G) = p$$ for alli, j ∈ C k and letting these probabilities be mutually independent. We show that forp=λ/k, λ>1,G k, p almost surely contains a connected component of sizec2 k ,c=c(λ). It is also true that the second largest component is of sizeo(2 k ).

Locating Central Paths in a Graph

Abstract: Concepts which measure the centrality of a vertex in a graph (eccentricity, distance and branch weight) are extended to paths in a graph. Locating paths with minimum eccentricity and distance, respectively, may be viewed as multicenter and multimedian problems, respectively, where the facilities are located on vertices that must constitute a path. The third problem is to find a path P in a graph for which the number of vertices in the largest component of G-P is minimized. The relationships among these concepts are studied. Most of the results presented are for trees, and, in particular, linear algorithms for finding paths in trees of minimum eccentricity and of minimum branch weight are presented. These problems arise in determining a “most accessible” linear route in a network according to several plausible criteria.

Distinguishing Vertices of Random Graphs

Abstract: The distance sequence of a vertex x of a graph is (d i (x)) n 1 where d i (x) is the number of vertices at distance i from x. The paper investigates under what condition it is true that almost every graph of a probability space is such that its vertices are uniquely determined by an initial segment of the distance sequence. In particular, it is shown that for r ≥ 3 and ɛ > O almost every labelled r-regular graph is such that every vertex x is uniquely determined by (d i (x)) u 1 , where u = ⌊ ( 1 2 + ɛ ) log n log ( r - 1 ) ⌋ . Furthermore, the paper contains an entirely combinatorial proof of a theorem of Wright [10] about the number of unlabelled graphs of a given size.

The maximal flow through a directed graph with random capacities

Abstract: We consider the maximal flow through a randomly capacitated network. We show that the maximal flow F n between vertices 0 and ∞ on of the complete graph on {0.1.2.…,n –1∞}. whose edges (i.j) are directed from i to j if and only if i E(B) almost surelv and in mean, as n→>∞: here E(B) is the mean value of a typical edge capacity. This answers a question posed by Grimmett and Welsh [I].

Orientations of circle graphs

Abstract: Let C(v1, …,vn) be a system consisting of a circle C with chords v1, …,vn on it having different endpoints. Define a graph G having vertex set V(G) = {v1, …,vn} and for which vertices vi and vj are adjacent in G if the chords vi and vj intersect. Such a graph will be called a circle graph. The chords divide the interior of C into a number of regions. We give a method which associates to each such region an orientation of the edges of G. For a given C(v1, …,vn) the number m of different orientations corresponding to it satisfies q + 1 ≤ m ≤ n + q + 1, where q is the number of edges in G. An oriented graph obtained from a diagram C(v1, …,vn) as above is called an oriented circle graph (OCG). We show that transitive orientations of permutation graphs are OCGs, and give a characterization of tournaments which are OCGs. When the region is a peripheral one, the orientation of G is acyclic. In this case we define a special orientation of the complement of G, and use this to develop an improved algorithm for finding a maximum independent set in G.

A shortness exponent for r‐regular r‐connected graphs

Abstract: Let r≧ 3 be an integer. It is shown that there exists e= e(r), 0 0 such that for all n ≧ N (if r is even) or for all even n ≧ N(if r is odd), there is an r-connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than ne vertices.

Polarities of G. Higman's symmetric design and a strongly regular graph on 176 vertices

Abstract: We investigate the polarities of G. Higman's symmetric 2-(176, 50, 14) design and find that there are two of them (up to conjugacy), one having 80 and the other 176 absolute points. From the latter we can derive a strongly regular graph with parameters (v, k, λ, μ)=(176, 49, 12, 14). Its group of automorphisms is Sym(8) with orbits of size 8 and 168 on the vertices. It does not carry a partial geometry or a delta space, and is not the result of mergingd=1 andd=2 in a distance regular graph with diameter 3 and girth 6 on 176 vertices.

On balance in group graphs

Abstract: We propose a generalization of signed graphs, called group graphs. These are graphs regarded as symmetric digraphs with a group element s(u, v) called the signing associated with each arc (u, v) such that s(u, v) s (v, u) = 1. A group graph is balanced if the product s(v1, v2) s (v2, v3) …s(um, v1) = 1 for each cycle v1, v2,…, vm, v1 in the graph. Let G denote the graph, F the group (not necessarily commutative), and s the signing. Then the group graph is denoted by (G, F, s). Given a group graph (G, F, s), which need not be balanced, we define the deletion index δ(G, F, s) as the minimum cardinality of a deletion set, which is a subset of lines whose deletion results in a balanced group graph. Similarly we define the alteration index γ(G, F, s) as the minimum cardinality of a alteration set, which is a set of lines {u, v} in the graph the values s(u, v) and s(v, u) of which can be changed so that the new group graph is balanced. When F is the group of order 2, we obtain a signed graph. Generalizing results known for signed graphs, we prove that minimal deletion sets are minimal change sets, which implies the equality δ(G, F, s) = γ(G, F, s) for all G, F, and s. We also prove the in-equality δ (G, F, s) ≤ (n - 1)q/n, where q is the number of lines in the graph G and n is the order of the group F. We conclude by studying δ for signed complete bigraphs Kn,n when the signing is determined from a Hadamard matrix.

Bridges of longest circuits and path coverings of labelled trees

Abstract: Let H be a subgraph of a graph G. We say that two edges el, e 2 E G, el, e 2 ~ H are equivalent iff e 1 = e 2 or there exists a path p with the end-edges e 1 and e 2 and no inner vertex of p belongs to H. A class of equivalent edges together with all vertices incident to edges of this class is called a bridge S of H in G. The vertices of S ffl H are called the attaching vertices of S; the other vertices of S are called inner vertices. In [4] I investigated bridges of maximal circuits and paths in graphs with a certain minimal valency. In this paper properties of maximal circuits, maximal paths, and maximal odd circuits are studied. K. HAUSCHILD, H. I-IERRE and W. I~AUTENBERG [2] proved in 1971: (1) Let S denote a bridge o[ a maximal circuit L of length l in a 2-connected graph G. If S contains at least one circuit, then each maximal circuit L' i n S has length l' ~ l -- 1.