A graph is bipartite if and only if it is 2-colorable. Again A graph is bipartite if and only if every edge belongs to an odd number of bonds, minimal subsets of edges whose removal increases the number of components of the graph.
I have tried it but not understand how to implement it?
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You can use the python driver for apache-age to load the graph and implement a function to check if it is bipartite or not.
Check out this article for further help.
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The statement you proved is not right because bipartite graph are only covered with 2 different colour that are not adjacent to each other.for e.g there are two colours red and blue red edge is connected to blue only 
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Apache AGE leverages the PostgreSQL ecosystem and allows you to utilize custom functions to implement additional functionality. In the case of determining the bipartiteness of a graph, you can write a custom function in PostgreSQL and then incorporate it into your AGE queries.
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You can use the following steps to determine the bipartiteness in the graph:
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I would advise you to create a custom function using PostgreSQL to find bipartite-ness in a graph.
I think, as of now AGE doesn't support this feature but, you can implement it using Postgres because the AGE backend is created using PostgreSQL.
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Apache AGE does not provide built-in automatic graph partioning, but you can integrate PostgreSQL which will allows you to implement various bipartitioning strategies at the PostgreSQL level.
Apache AGE is a graph database that uses PostgreSQL as its backend. This means that any functionality that is not provided out-of-the-box by AGE can be implemented in PostgreSQL using custom functions. In the case of determining bipartiteness of a graph, a custom function can be written in PostgreSQL and then used within an AGE query.