Abstract. For a real Hilbert space (H, ,

As a novel member of the class of vertex-degree-based topological indices, the so-called Sombor index was recently introduced by Gutman on the chemical graphs. In this paper, we present the minimum Sombor index for unicyclic graphs with the diameter D>_2.

Let U ⊆ R n (res. D ⊂ R n) be an open (res. a compact) subset, and let L be an elliptic operator defined on C 2 (U, R) (res. C 2 (D, R)). In the present paper, we are going to extend the maximum principle for the function f ∈ C 2 (U, R) (res. f ∈ C 2 (D, R)) satisfying the equation Lf = ε , where ε is a real everywhere nonzero continuous function on U (res. D). Finally, we obtain some applications in mathematics and physics.

Given a simple graph G = (V, E) with maximum degree ∆. Let (V 0 , V 1 , V 2 ) be an ordered partition of V , where2 + 1} is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of G, denoted by u StR (G), is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree T of order n ≥ 3 we prove the sharp bound u StR (T ) ≤ 8n 9 .