Ordering Unicyclic Graphs with a Fixed Girth by p-Sombor Indices

The p-Sombor index of a graphs G is deffned as, SOp(G) = X xy∈E(G) (d p (x) + d p (y)) 1 p , where d(x) represents the degree of vertex x in graph G. Our focus centers on exploring the p-Sombor index of unicyclic graphs, speciffcally addressing graphs with a predetermined girth. We determine the ffrst four smallest p-Sombor index and identifying the corresponding graphs that achieve these extremes.