A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. We characterize magic line graphs of general graphs and describe some class of supermagic line graphs of bipartite graphs.

In this paper we survey the behavior of various graph operations on the graph parameters clique-width and NLC-width. We give upper and lower bounds for the clique-width and NLC-width of the modi ed graphs in terms of the clique-width and NLC-width of the involved graphs. Therefor we consider the binary graph operations join, co-join, sum, di erence, products, corona, substitution, and 1-sum, an...

Let G be a simple connected molecular graph with vertex set V(G) and edge set E(G). One important modification of classical Zagreb index, called hyper Zagreb index HM(G) is defined as the sum of squares of the degree sum of the adjacent vertices, that is, sum of the terms 2 [ ( ) ( )] G G d u d v  over all the edges of G, where ( ) G d u denote the degree of the vetex u of G. In this paper, th...

The first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper we present the lower bound on M1 and M2 among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which th...

In this paper, we study the Minimum Sum Coloring (MSC) problem on P4-sparse graphs. In the MSC problem, we aim to assign natural numbers to vertices of a graph such that adjacent vertices get different numbers, and the sum of the numbers assigned to the vertices is minimum. Based in the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph, we show that...

A matched sum graph G1MG2 of two graphs G1 and G2 of the same order n is obtained by adding to the union (or sum) of G1 and G2 a set M of n independent edges which join vertices in V (G1) to vertices in V (G2). When G1 and G2 are isomorphic, G1MG2 is just a permutation graph. In this work we derive bounds for the k-restricted edge connectivity λ(k) of matched sum graphs G1MG2 for 2 ≤ k ≤ 5, and...

For a graph H with at most n vertices and a weighing of the edges of Kn with integers, we seek a copy of H in Kn whose weight is minimal, possibly even zero. Of a particular interest are the cases where H is a spanning subgraph (or an almost spanning subgraph) and the case where H is a fixed graph. In particular, we show that relatively balanced weighings of Kn with {−r, . . . , r} guarantee al...