// mathgrab lib
export default { foo: 'foo' }

// your code
import mathgraph from 'mathgraph';
mathgrab.foo

This imports the default export.

// mathgrab lib
export const someObject = { foo: 'foo' }

// your code
import { someObject } from 'mathgraph';
someObject.foo

This imports a named export.

// mathgrab lib
export const someObject = { foo: 'foo' }
export default { bar: 'bar' }

// your code
import mathgraph, { someObject } from 'mathgraph';
mathgrab.someObject.foo
mathgrab.default.bar

You can even combine these two to import both kinds of exports in one line.

// mathgrab lib
export const someObject = { foo: 'foo' }
export default { bar: 'bar' }

// your code
import * as mathgraph from 'mathgraph';
mathgrab.someObject.foo
mathgrab.default.bar

This imports all exports as a single object. The default export will be on the default property of this object.

const mathgraph = require('mathgraph');

This only works in node, and is functionally the same as import *, but it's a dynamic runtime require, rather than than a compile time import. That means that typescript can't use the types of the imported file. require is just a function, so it returns a value of type any, thereby missing out on all typing info.

Don't use require with typescript, you pretty always want to use import instead.

import mathgraph = require('mathgraph');

And lastly, this is invalid syntax. So don't do that.