Any change in that neighbor's lawn or ornamental plants will be an indictment. Discuss what you plan to do, why, and what precautions you'll take to make sure your neighbor's property is unaffected. If the pesticide's label recommends limiting childrens' or pets' access to your yard for awhile, discuss that point, too. Then take those precautions, because the potential for causing harm is real. If you happen to see your neighbor while you're spraying, also point out that you're doing just what you said you'd do.
Unexpected fertilizer can create zebra stripes near the property line.
Runoff and sprinklers that reach too far can cause leopard spots of green. The clippings, shredded leaves, and dirt your mower kicks up are all good things to apply to your own turf. But if your blower or side-discharging mower throws lawn debris onto your neighbor's property, chances are that will make a mess. If you're lucky, your neighbor will just have to sweep the problem away. If you're not lucky, your neighbor may remember that you recently applied a lawn weed killer that also can affect sprouting flowers and vegetables.