How to call a method in vue app from the vue component

You can execute root instance method like this: this.$root.methodName()

  Vue.component('todo-item', {
      template: '<li>This is a todo</li>',
      methods: {
        test: function() {
            this.$root.aNewFunction();
        }
      },
      mounted() {
        this.test();
      }
    })

    new Vue({
      el: '#app',
      template: '<todo-item></todo-item>',
      methods: {
        aNewFunction: function() {
            alert("inside");
        }
      }
    })

Another solution which I think it's more accorded with Vuejs architecture, it's to use events listener & emitter between child-component and its parent to make communications.

Check this simple fiddle as a vision

 Vue.component('todo-item', {
  template: '<li>This is a todo</li>',
  methods: {
    test: function() {
      console.log('Emmit this Event.');
      this.$emit('yourevent');
    }
  },
  created() {
    this.test();
  }
});

new Vue({
  el: '#vue-app',
  data: {
    'message': '',
  },
  methods: {
    aNewFunction(event) {
      console.log('yourevent Is Triggered!');
      this.message = 'Do Stuff...';
    },
  }
});