Avoid sending images to irrelevents in website

In some cases, I made a website that will auto generate some graphs depends to different accounts.

At first I try to generate a png with same file name, send the png and delete the png when server receipt any other get or post requests. However, considering one case, there is an access of person B just after the graph is auto generated that originally for person A. The graph may be deleted due to the same file name and becomes the graph for person B then send to both (in my understanding).

So, I try to generate different file names for the pngs generated for different accounts. However, other accounts although may not see the png on website, but they may still be able to get others' png through path. Is it possible to avoid sending the images to irrelevents in a website of auto generating png?

Sorry for my poor English. Many thanks!