For a connected graph G, the Wiener index, denoted by W (G), is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by ε(G), is the sum of the eccentricity of individual vertices. In [4], the authors posed a conjecture which states that given a graph G with at least three vertices, the difference between W (G) and ε(G) decreases when an edge is contracted and proved that the conjecture is true when e is a bridge. In this manuscript, we confirm that the conjecture is true for any connected graph G with at least three vertices irrespective of the nature of the edge chosen.
In this paper we prove a strong comparison principle for radially decreasing solutions u, v ∈ C 1,α 0 ( BR ) of the singular equationsHere we assume that 1 < p < 2, δ ∈ (0, 1) and f, g are continuous, radial functions such that 0 ≤ f ≤ g but f ≡ g in B R . For the case p > 2 a counterexample is provided where the strong comparison principle is violated. As an application of strong comparison principle, we prove a three solution theorem for p-Laplace equation and illustrate with an example.
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