Topological indices (TIs) have an important role in studying properties of molecules. A main problem in mathematical chemistry is finding extreme graphs with respect to a given TI. In this paper extremal graphs with respect to the modified first Zagreb connection index for trees in general and for trees with given number of pendants, for unicyclic graphs with or without a fixed girth and connected graphs are determined, using methods with higher degree of generality with respect to the transformation techniques usually used in such context. These graphs are relevant for chemical studies.
Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system.The results presented in this paper fall into two categories: in the first part of the paper we are concerned with the computational aspects of finding a minimum harmonious coloring and in the second part we determine the exact value of the harmonious chromatic number for some particular graphs and classes of graphs. More precisely, in the first part we show that finding a minimum harmonious coloring for arbitrary graphs is APX-hard, the natural greedy algorithm is a Ω( √ n)-approximation, and, moreover, we show a relationship between the vertex cover and the harmonious chromatic number. In the second part we determine the exact value of the harmonious chromatic number for all 3-regular planar graphs of diameter 3, some non-planar regular graphs and cycle-related graphs.
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