Lower and upper orientable strong diameters of graphs satisfying the Ore condition

AbstractLet D be a strong digraph. The strong distance between two vertices u and v in D, denoted by sdD(u,v), is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v. For a vertex v of D, the strong eccentricity se(v) is the strong distance between v and a vertex farthest from v. The minimum strong eccentricity among all vertices of D is the strong radius, denoted by srad(D), and the maximum strong eccentricity is the strong diameter, denoted by sdiam(D). The lower (resp. upper) orientable strong radius srad(G) (resp. SRAD(G)) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G. The lower (resp. upper) orientable strong diameter sdiam(G) (resp. SDIAM(G)) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G. In this work, we determine a bound of the lower orientable strong diameters and the bounds of the upper orientable strong diameters for graphs G=(V,E) satisfying the Ore condition (that is, σ2(G)=min{d(x)+d(y)|∀xy∉E(G)}≥n), in terms of girth g and order n of G

Lower and Upper Orientable Strong Radius and Diameter of Cartesian Product of Paths

强有向图d中任意两个点u,V的强距离Sd(u,V)定义为d中包含u和V的最小有向强子图duV的大小(弧的数目).d中一点u的强离心率SE(u)定义为u到其他顶点的强距离的最大值.强有向图d的强半径SrAd(d)(相应的强直径SdIAM(d))定义为d中所有顶点强离心率的最小值(相应的最大值).无向图g的最小定向强半径SrAd(g)(相应的最大定向强半径SrAd(g))定义为d中所有强定向的强半径的最小值(相应的最大值).无向图g的最小定向强直径SdIAM(g)(相应的最大定向强直径SdIAM(g))定义为d中所有强定向的强直径的最小值(相应的最大值).本文确定了路和路的笛卡尔积的最小定向强半径SrAd(PMxPn)和强直径的值SdIAM(PMxPn),给出了最大定向强半径SrAd(PMxPn)的界并提出关于最大定向强直径SdIAM(PMxPn)的一个猜想.For two vertices u and v in a strong digraph D,the strong distance sd(u,v)between u and v is the minimum size(the number of arcs)of a strong sub-digraph of D containing u and v.For a vertex v of D, the strong eccentricity se(v)is the strong distance between v and a vertex farthest from v.The strong radius srad(D)(resp.strong diameter sdiam(D))is the minimum(resp.maximum)strong eccentricity among the vertices of D.The lower(resp.upper)orientable strong radius srad(G)(resp.SRAD(G))of a graph G is the minimum(resp.maximum)strong radius over all strong orientations of G.The lower(resp.upper) orientable strong diameter sdiam(G)(resp.SDIAM(G))of a graph G is the minimum(resp.maximum) strong diameter over all strong orientations of G.In this paper,we determine the lower orientable strong radius and strong diameter of Cartesian product of paths,and give bounds on the upper orientable strong radius and a conjecture of the upper orientable strong diameter of Cartesian product of paths

Investigations in the semi-strong product of graphs and bootstrap percolation

The semi-strong product of graphs G and H is a way of forming a new graph from the graphs G and H. The vertex set of the semi-strong product is the Cartesian product of the vertex sets of G and H, V(G) x V(H). The edges of the semi-strong product are determined as follows: (g1,h1)(g2,h2) is an edge of the product whenever g1g2 is an edge of G and h1h2 is an edge of H or g1 = g2 and h1h2 is an edge of H. A natural subject for investigation is to determine properties of the semi-strong product in terms of those properties of its factors. We investigate distance, independence, matching, and domination in the semi-strong product Bootstrap Percolation is a process defined on a graph. We begin with an initial set of infected vertices. In each subsequent round, uninfected vertices become infected if they are adjacent to at least r infected vertices. Once infected, vertices remain infected. The parameter r is called the percolation threshold. When G is finite, the infection either stops at a proper subset of G or all of V(G) becomes infected. If all of V(G) eventually becomes infected, then we say that the infection percolates and we call the initial set of infected vertices a percolating set. The cardinality of a minimum percolating set of G with percolation threshold r is denoted m(G,r). We determine m(G,r) for certain Kneser graphs and bipartite Kneser graphs

Lower and upper orientable strong radius and strong diameter of complete k-partite graphs

For two vertices a and v in a strong digraph D, the strong distance sd(u, v) between a and v is the minimum size (the number of arcs) of a strong sub-digraph of D containing a and v. For a vertex v of D, the strong eccentricity se(v) is the strong distance between v and a vertex farthest from v. The strong radius srad(D) (resp. strong diameter sdiam(D)) is the minimum (resp. maximum) strong eccentricity among the vertices of D. The lower (resp. upper) orientable strong radius srad(G) (resp. SRAD(G)) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G. The lower (resp. upper) orientable strong diameter sdiam(G) (resp. SDIAM(G)) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G. In this paper, we determine the lower orientable strong radius and diameter of complete k-partite graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of complete k-partite graphs. We also find an error about the lower orientable strong diameter of complete bipartite graph K,, given in [Y.-L. Lai, F.-H. Chiang, C.-H. Lin, T.-C. Yu, Strong distance of complete bipartite graphs, The 19th Workshop on Combinatorial Mathematics and Computation Theory, 2002, pp. 12-16], and give a rigorous proof of a revised conclusion about sdiam(K-m,K-n). (c) 2006 Published by Elsevier B.V

SMARANDACHE MULTI-SPACE THEORY, Second Edition

We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multi-space came into being by purely logic. Another is the mathematical combinatorics motivated by a combinatorial speculation, i.e., a mathematical science can be reconstructed from or made by combinatorialization. Both of them contribute sciences for consistency of research with that human progress in 21st century