### A characterization of some q-multiplicative functions

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A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given.

Consider the linear congruence equation ${x}_{1}+...+{x}_{k}\equiv b\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}{n}^{s})$ for $b\in \mathbb{Z}$, $n,s\in \mathbb{N}$. Let ${(a,b)}_{s}$ denote the generalized gcd of $a$ and $b$ which is the largest ${l}^{s}$ with $l\in \mathbb{N}$ dividing $a$ and $b$ simultaneously. Let ${d}_{1},...,{d}_{\tau \left(n\right)}$ be all positive divisors of $n$. For each ${d}_{j}\mid n$, define ${\mathcal{C}}_{j,s}\left(n\right)=\{1\le x\le {n}^{s}:{(x,{n}^{s})}_{s}={d}_{j}^{s}\}$. K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on ${x}_{i}$. We generalize their result with generalized gcd restrictions on ${x}_{i}$ and prove that for the above linear congruence, the number of solutions...

Menon’s identity is a classical identity involving gcd sums and the Euler totient function $\phi $. A natural generalization of $\phi $ is the Klee’s function ${\Phi}_{s}$. We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).