We consider vertex colorings of graphs in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of the coloring is the sum of the costs incurred at each vertex. The cost chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum cost coloring of the graph. We show that the cost c...

Let d(G, k) be the number of pairs of vertices of a graph G that are at distance k, λ a real number, and Wλ(G) = ∑ k≥1 d(G, k)kλ. Wλ(G) is called the Wiener-type invariant of G associated to real number λ. In this paper, the Wiener-type invariants of some graph operations are computed. As immediate consequences, the formulae for reciprocal Wiener index, Harary index, hyperWiener index and Tratc...

If G is a connected graph, then HA(G) = ∑ u6=v(deg(u) + deg(v))/d(u, v) is the additively Harary index and HM (G) = ∑ u6=v deg(u) deg(v)/d(u, v) the multiplicatively Harary index of G. G is an apex tree if it contains a vertex x such that G − x is a tree and is a k-apex tree if k is the smallest integer for which there exists a k-set X ⊆ V (G) such that G − X is a tree. Upper and lower bounds o...

In this paper, we present sharp bounds for the Zagreb indices, Harary index and hyperWiener index of graphs with a given matching number, and we also completely determine the extremal graphs. © 2010 Elsevier Ltd. All rights reserved.

A graph whose edges are labeled either as positive or negative is called a signed graph. A signed graph is said to be net-regular if every vertex has constant net-degree k, namely, the difference between the number of positive and negative edges incident with a vertex. In this paper, we analyze some properties of co-regular signed graphs which are net-regular signed graphs with the underlying g...

Given two graphs G and H and a function f ⊂ V (G) × V (H), Hedetniemi [9] defined the function graph GfH by V (GfH) = V (G)∪ V (H) and E(GfH) = E(G) ∪ E(H) ∪ {uv|v = f(u)}. Whenever G ∼= H, the function graph was called a functigraph by Chen, Ferrero, Gera and Yi [7]. A function graph is a generalization of the α-permutation graph introduced by Chartrand and Harary [5]. The independence number ...

The diameter of a graph is an important factor for communication as it determines the maximum communication delay between any pair of processors in a network. Graham and Harary [N. Graham, F. Harary, Changing and unchanging the diameter of a hypercube, Discrete Applied Mathematics 37/38 (1992) 265–274] studied how the diameter of hypercubes can be affected by increasing and decreasing edges. Th...

In chemical graph theory, distance-degree-based topological indices are expressions of the form ∑ u6=v F (deg(u), deg(v)), d(u, v)), where F is a function, deg(u) the degree of u, and d(u, v) the distance between u and v. Setting F to be (deg(u) + deg(v))d(u, v), deg(u)deg(v)d(u, v), (deg(u)+deg(v))d(u, v)−1, and deg(u)deg(v)d(u, v)−1, we get the degree distance index DD, the Gutman index Gut, ...

A tripartite-circle drawing of a tripartite graph is in the plane, where each part vertex partition placed on one three disjoint circles, and edges do not cross circles. We present upper lower bounds minimum number crossings drawings $K_{m,n,p}$. In contrast to 1- 2-circle drawings, which may attain Harary-Hill bound, our results imply that balanced restricted 3-circle complete are optimal.

The Harary-Hill conjecture states that for every n > 0 the complete graph on n vertices Kn, the minimum number of crossings over all its possible drawings equals H(n) := 1 4 ⌊n 2 ⌋⌊n− 1 2 ⌋⌊n− 2 2 ⌋⌊n− 3 2 ⌋ . So far, the lower bound of the conjecture could only be verified for arbitrary drawings of Kn with n ≤ 12. In recent years, progress has been made in verifying the conjecture for certain ...