 Closed walk

A closed walk is a walk with equal endpoint vertices, $v_m=v_0$. Under our convention this includes the zero-length walk $(v)$; returning after traversing at least one edge requires $m>0$. Closure imposes no other restriction on repeated vertices or edges. In a directed graph, a closed directed walk must also follow every directed edge in its permitted direction.^[ref]

In the shared simple graph, $A,F,A$ is a length-two closed walk that immediately backtracks along $AF$. It is not a trail, since $AF$ is used twice, and it is not a graph cycle. The length-six closed walk $A,B,C,A,D,E,A$ is a trail, since its six edges are distinct, but is not a graph cycle: it returns to $A$ midway through the journey. The length-three closed walk $A,B,C,A$ is a graph cycle, with all vertices distinct apart from the required endpoint repetition. In a simple graph that is an undirected graph, a graph cycle has length at least three; the two-step backtrack is excluded.^[ref]

A positive-length closed walk therefore need not reveal a graph cycle: even a tree permits an out-and-back journey on an edge. A positive-length closed trail in a simple graph does contain a graph cycle. For a proof outline, choose a shortest positive-length consecutive portion of the trail whose endpoint vertices coincide. It cannot repeat an internal vertex, since that would give a shorter such portion; it cannot have length one or two because self-loops and repeated edges are excluded. It is consequently a graph cycle. Cutting it out leaves another closed trail, possibly of length zero, so repetition partitions the original edges into graph cycles. In our six-step example these are the two triangles in a graph. A closed trail using every edge would be an Euler circuit; here $AF$ is omitted.

Counting return journeys makes closed walks useful even when they repeat edges. For a finite simple graph with adjacency matrix $M$, the walk count from powers of an adjacency matrix gives $(M^k{)}_{uv}$ length-$k$ walks from $u$ to $v$, for every nonnegative integer $k$. A proof outline is that matrix multiplication appends one allowed step, summing over the previous endpoint; $M^0$ is the identity matrix, recording the unique zero-length walk at each vertex. ConsequentlyThe trace counts closed walks with a specified starting vertex and traversal order, not distinct graph cycles. These formulas use adjacency entries, not arbitrary travel costs in a weighted graph.^[ref]

In our example, $(M^2{)}_{AA}=5$: the five length-two closed walks are $A,B,A$, $A,C,A$, $A,D,A$, $A,E,A$, and $A,F,A$. Also $(M^3{)}_{AA}=4$, going round either triangle in a graph in either direction, and $\mathrm{tr}(M^3)=12$: each of the two triangles in a graph supplies three choices of starting vertex and two directions. These exact counts for the constructed graph illustrate why return counts and graph cycle counts differ.

A closed walk naturally models a delivery or inspection journey that returns to its depot. Repeated vertices let it combine several excursions; repeated edges let it serve a dead end and return. For an Euclidean travelling salesman tour, one can first construct a closed walk visiting all locations and then use the triangle inequality to shortcut repeated visits without increasing total length. That argument assumes direct travel between the retained locations is permitted, as in a complete graph weighted by Euclidean distance; an arbitrary road network need not contain those shortcut edges.