Shortest path algorithm in 0-1-unoriented graph
You should extract reading the data and the Dijkstra to their own method. It helps debugging. You don't have enough information to make a meaningful heuristic for A* anyway.

When filling the adjacency matrix you do it as if it is directional while your post says it's not:

for (size_t vert = 0; vert &lt; mEdges; ++vert) {
    cin &gt;&gt; edStart &gt;&gt; edEnd &gt;&gt; weight;
    ajList.at(edStart - 1).push_back(pair&lt;size_t, size_t&gt;(edEnd, weight));
    ajList.at(edEnd - 1).push_back(pair&lt;size_t, size_t&gt;(edStart, weight));
}


I suggest you use a priority queue of some sort; it lets you remove the current node in \$O(\log n)\$ instead of \$O(n)\$ (the first inner for loop in your code).