Transactions on Combinatorics
Volume:6 Issue: 4, Dec 2017

The regular graph of ideals of the commutative ring Rý, ýdenoted by Γreg(R)ý, ýis a graph whose vertexý ýset is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if either I contains a J-regular element or J contains an I-regular elementý. ýIn this paperý, ýit is proved that the radius of Γreg(R) equals 3ý. ýThe central vertices of Γreg(R)are determinedý, ýtooý.

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.

The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.

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.

In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected or k-connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.