I'm new in Vue and I have a problem with conditional rendering of a class managed by a child component and a page

When I load a page I have two sections, a component (navbar) and the page (in my case Home). By adding or not a specific class I would like the navbar in transparent, only when it is loaded on the Home page.

So I wrote this part in the navbar template:

<navbar class="normalclass" :class={transparent:istransparent}>Something</navbar>

JS:

export default {

    name: 'Navbar',
    data() {
        return {
            istransparent:false, // By default is false

        }

    }
}

In the views Home.vue I have instead inserted this code:

import { ref } from 'vue'
export default {
name: 'Home',
 
setup () {
    const istransparent = ref(true)
    return {
        istransparent
    }
}}
// I also tried to insert just this code but it doesn't change the result.
  setup () {
     return {
        istransparent:true
     }
  }

In App.vue I have this part of the template that manages the rendering of the application

<Navbar />

<router-view />

<Footer />

I didn't understand exactly where I am wrong and what would be the best solution to do what I need.

I ask those who have more experience than me for help.

########## EDIT #########

Ok, I solved the problem with a different approach.

Instead of using a variable that is to be read in individual components (or in this case view), such as home.vue

I used vue-router to find out what page I was on. So since I want the transparent header only in the home, in the Navbar.vue component I have added this code to say that if I am in home then the variable is true, so I add the classes I need. Here the code used:

In navbar.vue:

<navbar class="normalclass" :class= 
{transparent:isHome}>Something</navbar>

JS:

export default {

    name: 'Navbar',
        computed: {
            isHome() {
                 return this.$route.name == 'Home' // Check if router name is 'Home' 
        
            }
         }

    }
}

I don't know if that's the best approach, but it works and feels stable, with very little code. I share it with you for the sake of knowledge.