Recently two new degree concepts have been defined in graph theory: ev-degree and ve-degree. Also the evdegree and ve-degree Zagreb and Randić indices have been defined very recently as parallel of the classical definitions of Zagreb and Randić indices. It was shown that ev-degree and ve-degree topological indices can be used as possible tools in QSPR researches . In this paper we define the ve-degree and ev-degree Narumi-Katayama indices, investigate the predicting power of these novel indices and extremal graphs with respect to these novel topological indices. Also we give some basic mathematical properties of ev-degree and ve-degree NarumiKatayama and Zagreb indices.

A novel domination invariant defined by Boutrig and Chellali in the recent: total vertex-edge domination. In this paper we obtain an improved upper bound of total vertex edge-domination number of a tree. If is a connected tree with order , then ( ) ≤ 3 ⁄ with = 6⌈ 6 ⁄ ⌉ and we characterize the trees attaining this upper bound. Furthermore we provide a characterization of trees with ( ) = ( ).

The Merrifield-Simmons index σ = σ(G) of a graph G is the number of independent vertex sets of G. This index can be calculated recursively and expressed in terms of Fibonacci numbers. We determine the molecular graphs for which σ can be recursively calculated in a single step.

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