The Wiener number of a graph G is defined as 1 2 ∑ u,v∈V (G) d(u, v), d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, W (μ(S n)) ≤ W (μ(T k n )) ≤ W (μ(P k n )), where Sn, Tn and Pn denote a star, a gen...

The kth power of a graph G, denoted by Gk , is a graph with the same vertex set as G such that two vertices are adjacent in Gk if and only if their distance is at most k in G. The Wiener index is a distance-based topological index defined as the sum of distances between all pairs of vertices in a graph. In this note, we give the bounds on the Wiener index of the graph Gk . The Nordhaus–Gaddum-t...

A set D of vertices in a graph G = (V, E) is a dominating set of G if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. We define the cobondage number bc(G) of G to be the minimum cardinality among the sets of edges X ⊆ P2(V ) −E, where P2(V ) = {X ⊆ V : |X| = 2} such that γ(G + X) < γ(G). In this paper, the exa...

Let G = (V,E) be a graph. A function f : V → {−1, 1} is called a bad function of G if ∑ u∈NG(v) f(u) ≤ 1 for all v ∈ V , where NG(v) denotes the set of neighbors of v in G. The negative decision number of G, introduced in [12], is the maximum value of ∑ v∈V f(v) taken over all bad functions of G. In this paper, we present sharp upper bounds on the negative decision number of a graph in terms of...

The generalized k-connectivity κk(G) of a graph G was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity λk(G), recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized...

Let G be a simple graph with n vertices, and let G be its complement. Let δ(G) = δ and ∆(G) = ∆ be the minimum degree and the maximum degree of vertices of G, respectively. In this paper, some upper bounds for the Laplacian spectral radius of the Nordhaus-Gaddum type are obtained as follows: λ1(G) + λ1(G) ≤ 3n+∆− δ − 5 + √ 2(n+∆)2 + 2(δ + 1)2 − 8nδ 2 , λ1(G) + λ1(G) ≤ n+∆− δ − 1 + √( 2− 1 ω(G) ...

This paper shows the equivalence of various integer functions to the integer sequence A002620, and to the maximum of the product of certain pairs of combinatorial or graphical invariants. This maximum is the same as the upper bound of the Nordhaus-Gaddum inequality and related to Turán’s number. The computer algebra program MAPLE is used for solutions of linear recurrence and differential equat...

A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on vertices receiving the same colour is at most k. The k-defective chromatic number χk(G) of a graph G is the least positive integer m for which G is (m, k)colourable. The Nordhaus-Gaddum problem is to find sharp bounds for χk(G)+χk(G) and χk(G).χk(G) over the set of...

Given an integer k ≥ 2, we consider vertex colorings of graphs in which no k-star subgraph Sk = K1,k is polychromatic. Equivalently, in a star-[k]-coloring the closed neighborhood N[v] of each vertex v can have at most k different colors on its vertices. The maximum number of colors that can be used in a star-[k]-coloring of graph G is denoted by χ̄k⋆(G) and is termed the star-[k] upper chromati...

The minimum rank of a graph has been an interesting and well studied parameter 6 investigated by many researchers over the past decade or so. One of the many unresolved questions on 7 this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This 8 conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank 9 of its comp...