Clearest notation for vertices visited on shortest path I'm trying to describe the cost of the shortest path traversed on a graph $G(V, E)$. I'm wondering if the following notation below the summation is (1) clear and (2) professional.
$$
\min \sum_{\left\langle v_{1} \ldots v_{h} \ldots v_{H}\right\rangle} w_{v_{h}}^{\prime v_{h+1}}
$$
Namely, I am trying to convey the path length is a sum of edge weights between vertices visited by my path. Part of my confusion is finding a generalized way to express ordered pairs along a visited path; after reading other posts I cannot seem to find a clean way to express ordered pairs on a visited path (with out explicitly writing out: "$v   
 \epsilon$ path").
 A: The notation is not clear. The major problems are:

*

*It does not make sense to ask for the minimum value of an expression without saying what the variables are (which we can change to make the expression smaller or larger). A common notation is something like $\min \{f(x) : x\in S\}$: the minimum value of $f(x)$ over all $x \in S$.

*If $\langle v_1 \dots v_h \dots v_H\rangle$ is the only thing appearing below the sum, then you are saying "we are taking a sum of all possible objects of the form $\langle v_1 \dots v_h \dots v_H\rangle$". Instead, if $\langle v_1 \dots v_h \dots v_H\rangle$ were a fixed path, you'd probably just want a $\sum_{h=1}^{H-1}$.

*None of the variables appearing in the expression are defined at any point.

*The clearest way to say "the cost of the shortest path traversed on a graph $G(V,E)$" is to say "the cost of the shortest path traversed on a graph $G(V,E)$". The remaining questions left by that phrase are "Which paths are we considering? Is there a fixed starting point or ending point? Must we visit a specific set of vertices?" None of the questions are answered by the given expression.

As a minor note, writing ${w'}_{v_h}^{v_{h+1}}$ has nested subscripts, a subscript inside a superscript, and a superscript on a letter that already has $'$ attached to it. All three are to be avoided if possible.
Since I do not entirely understand the optimization problem you are trying to pose, I cannot give a complete description of the problem. But here is a starting point you might consider:

Let $w(e)$ denote the cost of traversing an edge $e$. For a path $P = \langle v_1, v_2, \dots, v_n\rangle$, its total cost $w(P)$ is defined to be the sum of the costs of its edges: $$w(P) := \sum_{i=1}^{n-1} w(v_i v_{i+1}).$$ Let $\mathscr P$ denote the set of all paths that (satisfy some condition). We are interested in finding the shortest cost of any such path: $\min \{w(P) : P \in \mathscr P\}$.

