In this paper we perform a further investigation for r-gcd-sum and r-lcm-sum functions. By making use of the properties generalization Euler’s φ-function, Abel’s identity elementary arguments, derive asymptotic formulas average function, function their generalizations. Moreover, also study sums reciprocals r-gcd r-lcm.

For a complete graph $$K_n$$ of order n, an edge-labeling $$c:E(K_n)\rightarrow \{ -1,1\}$$ satisfying $$c(E(K_n))=0$$ , and spanning forest F we consider the problem to minimize $$|c(E(F'))|$$ over all isomorphic copies $$F'$$ in . In particular, ask under which additional conditions there is zero-sum copy, that is, copy with $$c(E(F'))=0$$ We show always $$|c(E(F'))|\le \Delta (F)+1$$ where $...

A sum graph is a finite simple whose vertex set labeled with distinct positive integers such that two vertices are adjacent if and only the of their labels itself another label. The spum $G$ minimum difference between largest smallest in consisting number additional isolated necessary so labeling exists. We investigate various families graphs, namely cycles, paths, matchings. introduce sum-diam...