sagemathgh-36584: implemented power of graph function under basic met…

…hods

    
### Description

This PR introduces a new function, `power_of_graph`, to compute the kth
power of an undirected, unweighted graph efficiently using the shortest
distances method. The proposed function leverages Breadth-First Search
(BFS) to calculate the power graph, providing a practical and scalable
solution.

### Why is this change required?

The change is required to address the need for efficiently calculating
the kth power of a graph, a fundamental operation in graph theory. The
PR aims to incorporate this feature into the SageMath library.

Fixes sagemath#36582

### Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion (if applicable).
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.

### ⌛ Dependencies

sagemath#36582 : To implement power of graph
    
URL: sagemath#36584
Reported by: saatvikraoIITGN
Reviewer(s): David Coudert, saatvikraoIITGN

src/sage/graphs/generic_graph.py

@@ -15785,6 +15785,95 @@ def distance_all_pairs(self, by_weight=False, algorithm=None,
                                             weight_function=weight_function,
                                             check_weight=check_weight)[0]
 
+    def power(self, k):
+        r"""
+        Return the `k`-th power graph of ``self``.
+
+        In the `k`-th power graph of a graph `G`, there is an edge between
+        any pair of vertices at distance at most `k` in `G`, where the
+        distance is considered in the unweighted graph. In a directed graph,
+        there is an arc from a vertex `u` to a vertex `v` if there is a path
+        of length at most `k` in `G` from `u` to `v`.
+
+        INPUT:
+
+        - ``k`` -- integer; the maximum path length for considering edges in
+          the power graph.
+
+        OUTPUT:
+
+        - The kth power graph based on shortest distances between nodes.
+
+        EXAMPLES:
+
+        Testing on undirected graphs::
+
+            sage: G = Graph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)])
+            sage: PG = G.power(2)
+            sage: PG.edges(sort=True, labels=False)
+            [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (2, 5), (3, 4), (4, 5)]
+
+        Testing on directed graphs::
+
+            sage: G = DiGraph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)])
+            sage: PG = G.power(3)
+            sage: PG.edges(sort=True, labels=False)
+            [(0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 2), (1, 3), (1, 4), (1, 5), (2, 0), (2, 1), (2, 3), (2, 4), (2, 5), (3, 0), (3, 1), (3, 2), (4, 5)]
+
+        TESTS:
+
+        Testing when k < 0::
+
+            sage: G = Graph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)])
+            sage: PG = G.power(-2)
+            Traceback (most recent call last):
+            ...
+            ValueError: distance must be a non-negative integer, not -2
+
+        Testing when k = 0::
+
+            sage: G = Graph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)])
+            sage: PG = G.power(0)
+            sage: PG.edges(sort=True, labels=False)
+            []
+
+        Testing when k = 1::
+
+            sage: G = Graph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)])
+            sage: PG = G.power(1)
+            sage: PG.edges(sort=True, labels=False)
+            [(0, 1), (0, 3), (1, 2), (2, 3), (2, 4), (4, 5)]
+
+        Testing when k = Infinity::
+
+            sage: G = Graph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)])
+            sage: PG = G.power(Infinity)
+            Traceback (most recent call last):
+            ...
+            ValueError: distance must be a non-negative integer, not +Infinity
+
+        Testing on graph with multiple edges::
+
+            sage: G = DiGraph([(0, 1), (0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 5)], multiedges=True)
+            sage: PG = G.power(3)
+            sage: PG.edges(sort=True, labels=False)
+            [(0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 2), (1, 3), (1, 4), (1, 5), (2, 0), (2, 1), (2, 3), (2, 4), (2, 5), (3, 0), (3, 1), (3, 2), (4, 5)]
+        """
+        from sage.graphs.digraph import DiGraph
+        from sage.graphs.graph import Graph
+
+        power_of_graph = DiGraph() if self.is_directed() else Graph()
+
+        for u in self:
+            for v in self.breadth_first_search(u, distance=k):
+                if u != v:
+                    power_of_graph.add_edge(u, v)
+
+        if self.name():
+            power_of_graph.name("power({})".format(self.name()))
+
+        return power_of_graph
+
     def girth(self, certificate=False):
         """
         Return the girth of the graph.