THE SHANNON CAPACITY OF A GRAPH

Transcription

1 THE SHANNON CAPACITY OF A GRAPH JESPER M. MØLLER. Mathematics of communication An alphabet with k non-confusable letters can produce k r non-confusable words in r letters. Suppose now that we have letters B, P, T, D, and V. We may sometimes not be able to distinguish between B and P or B and V. We make a graph where confusable letters are connected by an edge: P T B D How can we best use these letters for error-free communication? We could use just B and T because no confusion is possible between these two letters, and just ignore the other letters. This would give us an error-free alphabet of letters. However, is this the most optimal use considering that we do not use three of the letters at all? Another possibility is to form words of two letters. The five -letter words (B, V ), (V, T ), (D, B), (T, D), (P, P ) can not be mixed up. If we receive (B, D) we can detect the error as (B, D) is not on the list. We can not correct the error because (B, D) could be a misinterpretation of both (B, V ) and (V, T ). These five -letter words work just as well as an error-free alphabet with k = letters. This is an improvement over the -letters B and T. V (B, P ) (V, P ) (D, P ) (B, T ) (V, T ) (D, T ) (P, D) (B, D) (V, D) (D, D) (P, V ) (B, V ) (V, V ) (P, B) (B, B) (V, B) Date: November,.

2 JESPER M. MØLLER Maybe we could improve the efficiency further by using 3-letter words? We now distill the ingredients of this real world problem into a problem in graph theory-. Graphs Definition (Finite graph). A graph G consists of a finite set V (G) and a set E(G) of -subsets of V (G). An element of V (G) is vertex of G. An element of E(G) is an edge of G. The adjacency matrix of G is the symmetric square matrix A G : V (G) V (G) {, } with A G (u, v) = if {u, v} E(G) and A G (u, v) = otherwise. The distance matrix of G is the symmetric square matrix D G : V (G) V (G) [, ] where D G (u, v) is minimum number of edges in a walk between u and v; in particular D G (u, u) =, and, by convention, D G (u, v) = inf = if u and v are not connected by edges. This distance function makes V (G) a finite metric space (with a metric that may take value ). The adjacency and distance matrices for the cyclic graph C on vertices are A C = D C = 3. Stability numbers and the Shannon capacity Definition (Stable set). A set A V (G) of vertices is stable if every edge of G contains at most one vertex from A. Definition 3 (Covering set). A set B V (G) of vertices is covering if every edge of G contains at least one vertex from B. Definition 4 (Stability number and covering number). Let G be a graph. The stability number of G is and the covering number of G is α(g) = max{ A A V (G) stable} β(g) = max{ B B V (G) covering} These numbers are complementary in the sense that α(g) + β(g) = V (G) (Gallai 99). Computation of α(g) is N P-hard. 6 7 α(g) = 4, β(g) =

3 THE SHANNON CAPACITY OF A GRAPH 3 α(c ) =, β(c ) = Definition (Strong product). The strong product of G and H is the graph G H with vertex set V (G) V (H). The vertices, (u, v ) and (u, v ) in V (G) V (H), are adjacent if and only if u and u are adjacent in G and v and v are adjacent in H. (Here, we take adjacent to mean identical or in the same edge.) As to stability numbers we clearly have that α(g H) α(g)α(h) since the product of a stable set of G with one of H is stable in G H. The strong product is denoted by the symbol because = The vertices adjacent to (, 3) in the strong product C C of the cyclic graph C with itself are (, 4) (, 4) (3, 4) (, 3) (, 3) (3, 3) (, ) (, ) (3, ) and the strong product C C is the graph on the torus 4 3 α(c C ) = 3 with the top and bottom horizontal edges and the left and right vertical edges identified. Definition 6 (Shannon capacity (Shannon 96)). [4] The Shannon capacity of a graph G is k Θ(G) = lim α(g k ) k 4

4 4 JESPER M. MØLLER This limit does exist and it is equal to sup k k α(g k ). The sequence k k α(g k ) is not monotone in general. For C this sequence begins with,,.... Θ(G) α(g) because α(g k ) α(g) k for all k. Computation of the Shannon capacity hard is difficult. The complexity class is unknown. Shannon himself computed the Shannon capacity for all graphs with at most 4 vertices. It took more than years before the capacity of the cyclic graph C was determined. Theorem 7 (Lovász 979). [3, ] Θ(C ) = The capacity of the cyclic graph C 7 on 7 vertices is unknown. 4. The magnitude of a graph The magnitude of a graph G is a power series k= M k(g)q k with integral coefficients M k (G) defined from the metric D G on the vertex set. Let Z[[q]] be the ring of power series in the variabel q with integral coefficients. Let q D G : V (G) V (G) Z[[q]] be symmetric square matrix over Z[[q]] whose (u, v)-entry is q D G (u, v) = q D G(u,v) (with the convention that q = ). Definition 8 (Weighting and magnitude of a graph). [] A weighting for G is a vector ω G : V (G) Z[[q]] such that q D G ω G =. The magnitude of G is the total weight v V (G) ω G(v) Z[[q]] of G. Example 9. For the cyclic graph C on five vertices q q ( + q + q q ) q C = q q q q q q, ω ( + q + q ) C = ( + q + q ) q q ( + q + q q ) ( + q + q ) and the magnitude M(C ) = +q+q = q + q + q 3 q 4 q +. Equivalently, the weighting of G is the function ω G such that u V (G): q DG(u,v) ω G (v) = u V (G) and the magnitude of G is the sum (q D G ) (u, v) u,v V (G) of all entries of the inverse to the matrix q D G. The adjugate matrix adj(q D G ): V (G) V (G) Z[q] takes (u, v) to the determinant of the matrix obtained from q D G by deleting row v and column u multiplied by a sign. Since (q D G )adj(q D G ) is the determinant times the identity matrix we see that u,v V (G) adj(qd G )(u, v) det(q D G ) is a rational function in q. In the special case where the row sums s(q) = v V (G) qd G(u,v) are independent of u, the weighting is constant ω G (v) = /s(q), and the magnitude of G is v V (G) ω G (v) = V (G) s(q)

5 THE SHANNON CAPACITY OF A GRAPH This happens for the cyclic graph C, for the Petersen graph P, and for the complete graph K n on n vertices. We conclude that these graphs have magnitudes M(C ) = + q + q = q q + q 3 q 4 q + 6q 6 + M(P ) = + 3q + 6q = 3q + 3q + 9q 3 4q 4 + 8q + 7q 6 + n M(K n ) = ( n)q = n( + ( n)q + ( n) q + ( n) 3 q 3 + ( n) 4 q 4 ) Definition (Graph product). The graph product of G and H is the graph G H with vertex set V (G) V (H). The vertices, (u, v ) and (u, v ) in V (G) V (H), are adjacent if and only if u = u and v and v are adjacent in H or v = v and u, u are adjacent in G. It is known that M(G H) = M(G)M(H), M (G) = V (G), and M (G) = E(G). But still M(G) remains a mystery. What does magnitude M(G) tell about the graph G? References. Willem Haemers, On some problems of Lovász concerning the Shannon capacity of a graph, IEEE Trans. Inform. Theory (979), no., 3 3. MR 37 (8g:944). Tom Leinster, The magnitude of metric spaces, Doc. Math. 8 (3), MR László Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory (979), no., 7. MR 496 (8g:9) 4. Claude E. Shannon, The zero error capacity of a noisy channel, Institute of Radio Engineers, Transactions on Information Theory, IT- (96), no. September, 8 9. MR 893 (9,63b) Institut for Matematiske Fag, Universitetsparken, DK København address: moller@math.ku.dk URL: htpp://