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Suppose G is a simple graph. The ℓ−eigenvalues δ 1 , δ 2 , . . . , δn of G are the eigenvalues of its normalized Laplacian ℓ. The normalized Laplacian Estrada index of the graph G is defined as ℓEE = ℓEE(G) = Σ n i=1 e δ i . In this paper the basic properties of ℓEE are investigated. Moreover, some lower and upper bounds for the normalized Laplacian Estrada index in terms of the number of vertices, edges and the Randic index are obtained. In addition, some relations between ℓEE and graph energy E ℓ (G) are presented.

ABSTRACT. The Graovac-Pisanski index (GP index) is an algebraic approach for generalizing the Wiener index. In this paper, we compute the difference between the Wiener and GP indices for an infinite family of polyhedral graphs.

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Let G be a graph with edge set E G and e = u v ∈ E G . Define n u e , G and m u e , G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v , respectively. The numbers n v e , G and m v e , G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as M o G = ∑ u v ∈ E G n u u v , G − n v u v , G and M o e G = ∑ u v ∈ E G m u u v , G − m v u v , G , respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.