894 research outputs found

    Wiener Index and Remoteness in Triangulations and Quadrangulations

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    Let GG be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If σ‾(v)\overline{\sigma}(v) denotes the arithmetic mean of the distances from vv to all other vertices of GG, then the remoteness of GG is defined as the largest value of σ‾(v)\overline{\sigma}(v) over all vertices vv of GG. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity

    Diameter of orientations of graphs with given order and number of blocks

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    A strong orientation of a graph GG is an assignment of a direction to each edge such that GG is strongly connected. The oriented diameter of GG is the smallest diameter among all strong orientations of GG. A block of GG is a maximal connected subgraph of GG that has no cut vertex. A block graph is a graph in which every block is a clique. We show that every bridgeless graph of order nn containing pp blocks has an oriented diameter of at most n−⌊p2⌋n-\lfloor \frac{p}{2} \rfloor. This bound is sharp for all nn and pp with p≥2p \geq 2. As a corollary, we obtain a sharp upper bound on the oriented diameter in terms of order and number of cut vertices. We also show that the oriented diameter of a bridgeless block graph of order nn is bounded above by ⌊3n4⌋\lfloor \frac{3n}{4} \rfloor if nn is even and ⌊3(n+1)4⌋\lfloor \frac{3(n+1)}{4} \rfloor if nn is odd.Comment: 15 pages, 2 figure
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