to my familyGracias a mis padres y hermanos por todo el apoyo y confianza desde que decidí emprender esta aventura, pero sobre todo por el infinito cariño y amor. A todos mis amigos que siempre han estado presentes de una u otra manera, gracias por ese gran apoyo moral y por sus palabras deánimo. La vida me ha dado tantos amigos que no acabaría de mencionarlos. Infinitas gracias. Conclusions Contributions of this work Open problems BibliographyNote that, if we consider a graph G whose edges have length equal to one and a graph G k obtained from G stretching out their edges until length k, then δ(G k ) = kδ(G). Therefore, all the results in this work can be generalized when the edges of the graph have length equal to k.

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In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we have inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices.

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<abstract><p>In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we get inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices.</p></abstract>