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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.

Silicate based inorganic materials are important for the synthesis of a new inorganic molecules in which the studies for ultrahigh proton conductivity and catalysis. Quantitative structure-property and structureactivity relationships of the silicate oxygen networks necessitate expressions for the molecular topological features of these networks. In QSPR/QSAR studies, physicochemical characteristics and molecular topological indices such as atom-bond connectivity (ABC), geometric-arithmetic (GA), harmonic (H) and sum-connectivity (χ) indices are used to model the physicochemical properties of chemical compounds and networks. These topological indices are based on the degrees of the vertices (atoms) of a connected graph. Recently, two novel degree concepts have been defined in graph theory; ev-degrees and ve-degrees. In this study by using the ve-degree concept, we define ve-degree atom-bond connectivity (ve-ABC), ve-degree geometric-arithmetic (ve-GA), ve-degree harmonic (ve-H) and ve-degree sum-connectivity (ve-χ) indices as parallel to their corresponding classical degree versions. We show that the ve-degree sum-connectivity index give better correlation than Wiener, Zagreb and Randić indices to predict the acentric factor of octanes. Also, we compute the ve-degree topological indices for some silicate oxygen netwoks such as dominating oxide network (DOX), regular triangulene oxide network (RTOX), dominating silicate network (DSL) and derive analytical closed formulae of these networks.

Topological indices have important role in theoretical chemistry for QSPR researches. Among the all topological indices the Randić index has been used more considerably than any other topological indices in chemical and mathematical literature. Most of the topological indices as in the Randić index are based on the degrees of the vertices of a connected graph. Recently novel two degree concepts have been defined in graph theory; ev-degrees and ve-degrees. In this study ev-degree Randić index is defined by using ev-degree concept as parallel to their corresponding classical degree version. This new ev-degree Randić index is compared with the Randić index by modelling some physicochemical properties of octane isomers. It is showed that the evdegree Randić index give better correlation than the Randić index to predict the entropy, acentric factor and standard enthalpy of vaporization of octanes. Also the exact values of the ev-degree Randić index for the well-known graph classes such as; paths, cycles, stars and complete graphs are given.

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Topological indices are important tools to modeling chemical properties of molecules. ve-degree based atom-bond connectivity, sum-connectivity, geometric-arithmetic, and harmonic indices are defined as their corresponding classical degree based counterparts recently in chemical graph theory. In this study we investigate ve-degree atom-bond connectivity, sum-connectivity, geometric-arithmetic, and harmonic topological properties of copper oxide. We calculate exact values of these novel topological indices for copper oxide and give closed formulas. KEYWORDS Ve-degree atom-bond connectivity index; ve-degree sumconnectivity index; ve-degree geometric arithmetic index; vedegree harmonic index.

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