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# New bounds of extended energy of a graph

Dec 2022

The extended adjacency matrix $A_{ex}(G)$ of a graph $G$ with $n$ vertices isa real symmetric marix of order $n\times n$ whose $(i,j)$-th entry is theaverage of the ratio of the degree of the $i$-th vertex to that of the $j$-thvertex and its reciprocal, if the vertices $i$ and $j$ are adjacent, and 0,otherwise. Sum of absolute eigenvalues of the matrix $A_{ex}(G)$ is known asthe extended energy of $G$. In this paper, we introduce the notion of extended vertex energy and obtainupper and lower bounds of the same. Using the upper bounds of the extendedvertex energy, we obtain new upper bounds of the extended energy of a graph.Next, we obtain two inequalities which relate the extended energy with theordinary energy of a graph. One of those inequalities resolves a conjecturewhich states that the extended energy of any graph is not less than itsordinary energy. Using the relationships of extended energy and ordinaryenergy, we obtain new bounds of extended energy in terms of number of vertices,number of edges, minimum degree and maximum degree of the graph. We show thatthese new bounds are improvements of some existing bounds. Finally, someimproved Nordhaus-Gaddum-type bounds for extended energy are also obtained.

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