Abstract
We give sharp lower bounds for the Zagreb eccentricity indices of connected graphs with fixed numbers of vertices and edges, sharp lower and upper bounds for the Zagreb eccentricity indices of trees with fixed number of pendant vertices, sharp upper bounds for the Zagreb eccentricity indices of trees with fixed matching number (fixed maximum degree, respectively), and characterize the extremal graphs.
Highlights
Let G be a connected graph with vertex set V (G) and edge set E(G)
Let us point to the timely paper that just appeared in this journal by Stevanović[6] reporting on the relationship between these two descriptors, a topic that is in recent years considerably studied, e.g., in References 7 and 8
The Zagreb eccentricity indices were introduced in an analogous way as the Zagreb indices by Vukičević and Graovac.[11]
Summary
Let G be a connected graph with vertex set V (G) and edge set E(G). For denotes the degree of a u vinertGex.1,2uT heVf(iGrs)t,ZadgGre(bu ) or du index ofG is defined as:[3,4] M1(G) du[2 ], u V (G )while the second Zagreb index of G is defined as:[3,4] M2 (G) du dv . uv E (G)The properties of these molecular indices and their derivatives are continuously studied, e.g., in Reference 5. Let G be a connected graph with vertex set V (G) and edge set E(G). For a vertex u V (G), eG (u) or eu denotes the eccentricity of u in G, which is equal to the largest distance from u to other vertices.[1,2] The Zagreb eccentricity indices were introduced in an analogous way as the Zagreb indices by Vukičević and Graovac.[11] The first Zagreb eccentricity index of G is defined as:
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