The relation between the Wiener index W(G) and eccentricity $$\varepsilon (G)$$ of a graph G is studied. Lower upper bounds on in terms are proved extremal graphs characterized. A Nordhaus–Gaddum type result involving given. sharp bound tree its proved. It shown that class trees same order, difference $$W(T) - \varepsilon (T)$$ minimized caterpillars. An exact formula for radius T obtained. low...

Nordhaus-Gaddum results for weakly convex domination number of a graph G are studied.

We prove that, for every n-vertex graph G, the treewidth of G plus the treewidth of the complement of G is at least n − 2. This bound is tight. © 2012 Gwenaël Joret and David R. Wood. Published by Elsevier Ltd. All rights reserved. Nordhaus–Gaddum-type theorems establish bounds on f (G) + f (G) for some graph parameter f , where G is the complement of a graph G. The literature has numerous exam...

In this paper, we extend the recently introduced vertex-degree-based topological index, Sombor and call it general index. The index generalizes both forgotten We present bounds in terms of other important graph parameters for also explore Nordhaus–Gaddum-type result further relations between generalized indices: Randić sum-connectivity

Let G be a graph with n vertices and m edges and let μ1 (G) ≥ ... ≥ μn (G) be the eigenvalues of its adjacency matrix. We discuss the following general problem. For k fixed and n large, find or estimate fk (n) = max v(G)=n |μk (G)|+ ∣

Let χ(G) and χf (G) denote the chromatic and fractional chromatic numbers of a graph G, and let (n+, n0, n−) denote the inertia of G. We prove that: 1 + max ( n+ n− , n− n+ ) 6 χ(G) and conjecture that 1 + max ( n+ n− , n− n+ ) 6 χf (G). We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with...