Search

The Randić index R(G) of a graph G is the sum of the weights (dudv)1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randić indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex unicyclic graphs with the third maximum for n ≥ 5, the fourth maximum for n ≥ 7 and the fifth maximum for n ≥ 8. Recently, Li et al. [The Randić indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex unicyclic graphs with the sixth maximum and the seventh maximum for n ≥ 9 and the eighth maximum for n ≥ 10. In this paper, we characterize the n-vertex unicyclic graphs with the ninth maximum, the tenth maximum, the eleventh maximum, the twelfth maximum and the thirteenth maximum of Randić values. [ABSTRACT FROM AUTHOR]

The annihilating-ideal graph of a commutative ring R with unity is deffned as the graph AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect. [ABSTRACT FROM AUTHOR]

Let G be a fuzzy graph of order n, where(u) is the vertex membership, (u; v) is membership value of an edge and (u) is the strength of vertex. The first fuzzy Zagreb index is the sum -(ui)-(ui) + -(uj)-(uj) where uiuj 2 - and the corresponding fuzzy Zagreb matrix is the square matrix of order n whose (i; j)th entry whenever i 6= j, is -(ui)-(ui)+-(uj)-(uj) and zero otherwise. In this paper, we introduce the Zagreb Estrada index of fuzzy graphs and establish some bounds for it. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) and edge set E(G). The two Zagreb indices M1 =P v2V (G) d² G(v) and M2 = Puv2E(G) dG(u)dG(v) are vertex degree based graph invariants that have been introduced in the 1970s and extensively studied ever since. In this paper, we first give a lower bound on the first Zagreb index of trees with given Roman domination number and we characterize all extremal trees. Then we present upper bound for Zagreb indices of unicyclic and bicyclic graphs with given Roman domination number. [ABSTRACT FROM AUTHOR]

The annihilating-ideal graph of a commutative ring R with unity is deffned as the graph AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et.al. proved that AG(Zn) is weakly perfect. In this short paper, we characterize n for which AG(Zn) is perfect. [ABSTRACT FROM AUTHOR]