منابع مشابه
On diameter and inverse degree of chemical graphs
The inverse degree r(G) of a finite graph G = (V,E) is defined as r(G) = ∑ v∈V 1 d(v) , where d(v) is the degree of vertex v. In Discrete Math. 310 (2010), 940-946, Mukwembi posed the following conjecture: Let G be a connected chemical graph with diameter diam(G) and inverse degree r(G). Then diam(G) ≤ 12 5 r(G) +O(1). In this paper, we settle the conjecture affirmatively.
متن کاملOn diameter and inverse degree of a graph
The inverse degree r(G) of a finite graph G = (V, E) is defined as r(G) = v∈V 1 deg v , where deg v is the degree of vertex v. We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs.
متن کاملRadius, diameter, and minimum degree
We give asymptotically sharp upper bounds for the maximum diameter and radius of (i) a connected graph, {ii) a connected trangle-free graph, (iii) a connected C,-free graph with n vertices and with minimum degree 6, where n tends to infinity. Some conjectures for J&-free graphs are also stated. Let G be a connected graph with vertex set V(G) and edge set E(G). For any XJJ E V(G) let dG(x,y) den...
متن کاملAn inequality between the diameter and the inverse dual degree of a tree
Let R(T ), D(T ) be respectively the radius and diameter of a nontrivial tree T and I(T ) = ∑ u∈V (T ) 1/d(u) be the inverse dual degree, where d(u) = ( ∑ v∈N(u) d(v))/d(u) for each u ∈ V (T ). In this note we prove that I(T ) ≥ { R(T ) + 1/3, if D(T ) is odd R(T ) + 5/6, if D(T ) is even, with equality if and only if T is a path of at least 4 vertices. This inequality strengthens a conjecture ...
متن کاملDiameter Two Graphs of Minimum Order with Given Degree Set
The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2008
ISSN: 0012-365X
DOI: 10.1016/j.disc.2007.07.053