This is an answer to the question: "Given a $C^\ast$-algebra $A$ that shares some properties of (higher-rank) graph algebras (such as separability, nuclearity, UCT, K-theoretic properties), can I conclude that $A$ is a (higher-rank) graph algebra?"

My answer is: this would require a classification of the class of $C^\ast$-algebras in question together with a description of the range of the classifying invariant on (higher-rank) graph algebras. This problem is open, even for purely infinite Cuntz--Krieger algebras; see DO PHANTOM CUNTZ-KRIEGER ALGEBRAS EXIST?

In the simple purely infinite case, one can say the following: a unital UCT Kirchberg algebra $A$ is a graph algebra if and only if $K_1(A)$ is free. This theorem is due to Wojciech Szymanski.

This is an answer to the question: "Given a $C^\ast$-algebra $A$ that shares some properties of (higher-rank) graph algebras (such as separability, nuclearity, UCT, K-theoretic properties), can I conclude that $A$ is a (higher-rank) graph algebra?"

My answer is: this would require a classification of the class of $C^\ast$-algebras in question together with a description of the range of the classifying invariant on (higher-rank) graph algebras. This problem is open, even for purely infinite Cuntz--Krieger algebras; see Do phantom Cuntz-Krieger algebras exist? (Edit: this case was solved in http://arxiv.org/abs/1511.09463.)

In the simple purely infinite case, one can say the following: a unital UCT Kirchberg algebra $A$ is a graph algebra if and only if $K_1(A)$ is free. This theorem is due to Wojciech Szymanski.

This is an answer to the question: "Given a $C^\ast$-algebra $A$ that shares some properties of (higher-rank) graph algebras (such as separability, nuclearity, UCT, K-theoretic properties), can I conclude that $A$ is a (higher-rank) graph algebra?"

My answer is: this would require a classification of the class of $C^\ast$-algebras in question together with a description of the range of the classifying invariant on (higher-rank) graph algebras. This problem is open, even for purely infinite Cuntz--Krieger algebras; see DO PHANTOM CUNTZ-KRIEGER ALGEBRAS EXIST?

This is an answer to the question: "Given a $C^\ast$-algebra $A$ that shares some properties of (higher-rank) graph algebras (such as separability, nuclearity, UCT, K-theoretic properties), can I conclude that $A$ is a (higher-rank) graph algebra?"

My answer is: this would require a classification of the class of $C^\ast$-algebras in question together with a description of the range of the classifying invariant on (higher-rank) graph algebras. This problem is open, even for purely infinite Cuntz--Krieger algebras; see DO PHANTOM CUNTZ-KRIEGER ALGEBRAS EXIST?

In the simple purely infinite case, one can say the following: a unital UCT Kirchberg algebra $A$ is a graph algebra if and only if $K_1(A)$ is free. This theorem is due to Wojciech Szymanski.