جستجوی مقالات مرتبط با کلیدواژه « ?harmonic index? » در نشریات گروه « ریاضی »

The harmonic index of a graph G is defined as H(G)=∑uv∈E(G)2d(u)(v)ý, ýwhere d(u) denotes the degree of a vertex u in Gý. ýLet G(n,k) be the set of simple n-vertex graphs with minimum degree at least ký. ýIn this work we consider the problem of determining the minimum value of theý ýharmonic index and the corresponding extremal graphs among G(n,k)ý. ýWe solve the problem for each integer k(1≤k≤n/2) and show the corresponding extremal graph is the complete split graph K∗k,n−ký. ýThis result together with our previous result which solve the problem for each integer k(n/2≤k≤n−1) give a complete solution of the problemý.

The eccentricity of a vertex is the maximum distance from it toý ýanother vertex and the average eccentricity ecc(G) of aý ýgraph G is the mean value of eccentricities of all vertices ofý ýGý. ýThe harmonic index H(G) of a graph G is definedý ýas the sum of 2di over all edges vivj ofý ýGý, ýwhere di denotes the degree of a vertex vi in Gý. ýIný ýthis paperý, ýwe determine the unique tree with minimum averageý ýeccentricity among the set of trees with given number of pendentý ývertices and determine the unique tree with maximum averageý ýeccentricity among the set of n-vertex trees with two adjacentý ývertices of maximum degree Δý, ýwhere n≥2Δý. ýAlsoý, ýweý ýgive some relations between the average eccentricityý, ýthe harmonicý ýindex and the largest signless Laplacian eigenvalueý, ýand strengthený ýa result on the Randi'{c} index and the largest signless Laplacianý ýeigenvalue conjectured by Hansen and Lucas \cite{hl}ý.

ýThe harmonic index of a graph G is defined as the sum of the weightsý ý2degG(u)痨(v) of all edges uvý ýof Gý, ýwhere degG(u) denotes the degree of a vertex u in Gý. ýIn this paperý, ýwe study the harmonic index of subdivision graphsý, ýt-subdivision graphs and alsoý, ýS-sum and St -sum of graphsý.

ýThe harmonic index of a connected graph Gý, ýdenoted by H(G)ý, ýisý ýdefined as H(G)=∑uv∈E(G)2du ý ýwhere dv is the degree of a vertex v in Gý. ýIn this paperý, ýexpressions for the Harary indices of theý ýjoiný, ýcorona productý, ýCartesian productý, ýcomposition and symmetric difference of graphs areý ýderivedý.