678,207 research outputs found
Illumination by Taylor Polynomials
Let f(x) be a differentiable function on the real line R, and let P be a
point not on the graph of f(x). Define the illumination index of P to be the
number of distinct tangents to the graph of f which pass thru P. We prove that
if f '' is continuous and nonnegative on R, f '' > m >0 outside a closed
interval of R, and f '' has finitely many zeroes on R, then every point below
the graph of f has illumination index 2. This result fails in general if f ''
is not bounded away from 0 on R. Also, if f '' has finitely many zeroes and f
'' is not nonnnegative on R, then some point below the graph has illumination
index not equal to 2. Finally, we generalize our results to illumination by odd
order Taylor polynomials.Comment: Minor modifications and correction
A proof for a conjecture on the Randić index of graphs with diameter
AbstractThe Randić index R(G) of a graph G is defined by R(G)=∑uv1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n≥3 vertices with the Randić index R(G) and the diameter D(G), R(G)−D(G)≥2−n+12andR(G)D(G)≥n−3+222n−2, with equalities if and only if G is a path. In this work, we show that this conjecture is true for trees. Furthermore, we prove that for any connected graph on n≥3 vertices with the Randić index R(G) and the diameter D(G), R(G)−D(G)≥2−n+12, with equality if and only if G is a path
The Randic index and the diameter of graphs
The {\it Randi\'c index} of a graph is defined as the sum of
1/\sqrt{d_ud_v} over all edges of , where and are the
degrees of vertices and respectively. Let be the diameter of
when is connected. Aouchiche-Hansen-Zheng conjectured that among all
connected graphs on vertices the path achieves the minimum values
for both and . We prove this conjecture completely. In
fact, we prove a stronger theorem: If is a connected graph, then
, with equality if and only if is a path
with at least three vertices.Comment: 17 pages, accepted by Discrete Mathematic
The median function on graphs with bounded profiles
AbstractThe median of a profile π=(u1,…,uk) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of π. It is shown that for profiles π with diameter θ the median set can be computed within an isometric subgraph of G that contains a vertex x of π and the r-ball around x, where r>2θ−1−2θ/|π|. The median index of a graph and r-joins of graphs are introduced and it is shown that r-joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles
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