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Abstract

In 2018, Arizmendi and Juarez introduced the concept of energy of a vertex, a novel approach allowing the total energy of a graph to be expressed as the sum of the energies of its individual vertices. In this article, we extend the notion of energy of a vertex to the context of the Randi\'c matrix. We define the Randi\'c energy of a vertex and explore its mathematical properties through various combinatorial techniques. We derive several upper and lower bounds for the Randi\'c energy of a vertex. Furthermore, we establish that among the connected graphs, the central vertex of a star attains the maximum Randi\'c energy, whereas pendent vertices attain the minimum. Also, we report the Coulson-type integral formula for the Randi\'c energy of a vertex and its applications.