The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccen...The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.展开更多
Given a distribution of pebbles on the vertices of a connected graph G,a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex.The t-pebbling number f_(t)(G)of a simple...Given a distribution of pebbles on the vertices of a connected graph G,a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex.The t-pebbling number f_(t)(G)of a simple connected graph G is the smallest positive integer such that for every distribution of fteGT pebbles on the vertices of G,we can move t pebbles to any target vertex by a sequence of pebbling moves.Graham conjectured that for any connected graphs G and H,f_(1)(G×H)≤f1(G)f1(H).Herscovici further conjectured that fst(G×H)≤6 fseGTfteHT for any positive integers s and t.Wang et al.(Discret Math,309:3431–3435,2009)proved that Graham’s conjecture holds when G is a thorn graph of a complete graph and H is a graph having the 2-pebbling property.In this paper,we further show that Herscovici’s conjecture is true when G is a thorn graph of a complete graph and H is a graph having the 2t-pebbling property.展开更多
文摘The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.
文摘Given a distribution of pebbles on the vertices of a connected graph G,a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex.The t-pebbling number f_(t)(G)of a simple connected graph G is the smallest positive integer such that for every distribution of fteGT pebbles on the vertices of G,we can move t pebbles to any target vertex by a sequence of pebbling moves.Graham conjectured that for any connected graphs G and H,f_(1)(G×H)≤f1(G)f1(H).Herscovici further conjectured that fst(G×H)≤6 fseGTfteHT for any positive integers s and t.Wang et al.(Discret Math,309:3431–3435,2009)proved that Graham’s conjecture holds when G is a thorn graph of a complete graph and H is a graph having the 2-pebbling property.In this paper,we further show that Herscovici’s conjecture is true when G is a thorn graph of a complete graph and H is a graph having the 2t-pebbling property.
