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On generic forms of complementary graphs
On Generic Forms of Complementary Graphs by
J. W. MOON
University of Alberta, Edmonton, Alberta, Canada T6G 2Gl and S. D.
BEDROSIAN
Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. ABSTRACT : Each graph may be associated with a certainfunction called its genericform. If one knows the generic forms of given graphs, then one can easily determine the number of spanning trees in graphs obtained from a complete multi-graph either (1) by adding, or (2) by deleting the edges of disjoint copies of the given graphs. Our objective here is to give a proof of a simple and useful relation between the generic forms of graphs that are complementary with respect to a complete multi-graph.
I. Introduction
All graphs considered here have vertices labelled 1, 2,. . .,n unless indicated otherwise; there may be several edges joining the same pair of vertices, but we shall ignore the presence of loops. For given graphs A and B, let A @ B denote the graph obtained by identifying the vertices (but not the edges) of A and B in the obvious way ; if B is a subgraph of A, then A 0 B denotes the complement of B with respect to A. For any positive integer k, let kK denote the complete multigraph (on n vertices) in which each pair of distinct vertices is joined by K edges. For any given graph G, let c(G) denote the number of different spanning trees in G, let e(G) denote the total number of edges in G, let l(G) denote the number of connected components of G, and let p(G) denote the product of the number of vertices in the components of G. The genericformf(G,x) of the graph G may be defined by the relation
f(G 4 = 1
P(F) (-
W4@
FGG
where the sum is over all spanning forests F of G; notice that F(G, x) is a polynomial in l/x of degree n - l(G). (We remark that x”- tf(G, -x) is the same as the function B:(G) defined by Kelmans (6) as a certain determinant ; the main step in showing the equivalence between the two definitions is implicit in (13; p. 7); see also (8; p. 203) and (4; p. 38).) Generic forms can be used to determine the number of spanning trees in certain graphs by appealing to the following results, valid for any graph G and any positive integer k, provided that H is a subgraph of kK in Theorem II. Theorem I. c(kK @ H) = k(kn)“-‘f(H, Theorem
OThe
II. c(kK 0 II) = k(kn)“-2f(II,
Franklin1nst1tute00164032/83$3.00+0.00
- kn). kn).
187
J. W Moon and S. D. Bedrosian The case k = 1 of Theorem II is implicit in Temperley (13); but it seems that Bedrosian (1) was the first to discuss both results in this generality (6). For the sake of completeness we shall give short proofs of these results in Section II ; our proofs are closely related to the proof given in Ref. (11) for the case k = 1 of Theorem II which is perhaps the case most frequently applied in practice. In order to exploit Theorems I and II one needs, of course, to know the generic forms of the graphs H that arise. Formulas for the generic forms of various graphs may be found, for example, in (7), (12; Section 6.4) and (4; Section 7.6). There is a simple relation between the generic forms of complementary graphs, so that if one knows the generic form of a graph H then one can easily determine the generic form of the complementary graph kK @ H. This relation may be formulated as follows : Main result. If H is a subgraph of kK, then xn-‘f(kK
0 H,x) = (x-kn)“-‘f(H,kn-x).
(4
Relation (2) was discovered by Bedrosian and illustrated in (1); further applications were given in (2,3,9). Our primary objective here is to give a short proof of this relation in Section III that is based on Theorems I and II. An alternative approach to a relation equivalent to (2), based on determinants, is given in (6; pp. 2125-2126); see also (4; p. 58).
II. Proof of Theorems For any forest F and any graph G, let t(F, G) denote the number such that F @ J is a tree. It is known (l&12) that
of forests J in G
t(F, K) = p(F)n”-’ -@). Since the forests J must have n- 1 -e(F)
edges, it follows that
t(F,kK) = k”-‘-“‘F’t(F,K)
= kp(F)(kn)“-2p”(F)
(3)
for any positive integer k. (These results hold even if F has no edges.) We now count the number of spanning trees in the graph kK OH. It is not difficult to see that for each spanning forest F in H, there are t(F, kK) spanning trees T in kK @ H such that T n H = F. It follows, therefore, that c(kK @ H) = 1
t(F, kK) = k(kn)“p2 c
=
k(kn)“-2f(H,
p(F)(kn)-“‘F’
FEH
FEH
- kn)
as required, by (3) and (1). To count the spanning trees in kK 0 H, we must exclude from the set of all spanning trees in kK those which contain any edges of H. It is not difficult to see that for each spanning forest F in H, there are t(F, kK) spanning trees Tin kK such that F 5 T n H. It follows, therefore, by the method of inclusion and exclusion (5; p. 96) that c(kK @ H), the number of spanning trees T in kK containing no edges of H, is 188
Journal of the Franklin Institute Press Ltd.
rergamon
On Generic Forms of Complementary
Graphs
given by the formula c(kK @ H) = =
1 (- l)“%(F, kK) = k(kn)“-’ FEH k(kn)“-‘f(H:
1
p(F)( - kn)-“cF)
FEH
kn),
as required.
III. Proof of Main Result Consider
the graph G=gK@H=(g+k)K@{kKOH}
(4)
when g is an arbitrary positive integer. If we apply Theorems second expressions for G, we find that c(G) = s(s$
- 2f(K
I and II to the first and
- gn)
(5)
and c(G) = (g + k) {(g + k)njnP2f(kK
0 H, (g + k)n).
(6)
Now consider the functions z”- ’f(H, -z) and (z + kn)“- ‘f(kK 0 H, z + kn), for fixed H, n, and k. Both functions are polynomials in z of degree n - 1. It follows from (5) and (6) that these polynomials are numerically equal when z = gn for g = 1,2,. . . ; hence, they must be identically equal for all z, i.e. z”-‘f(H,
-z)
= (z+kn)“-‘f(kKOH,z+kn).
If we let x = z + kn, this can be rewritten xn-lf(kK This completes
(7)
as
0 H,x) = (x-kn)“-‘f(H,
kn-x).
(8)
the proof of the main result.
IV. Concluding Remarks If we apply relation
(7) twice to the graph G defined in (4), we find that
(z-gn)“-‘f(ghOH,gn-z)=(z+kn)“-‘f(kKOH,kn+z)=z”-‘f(H,-z); hence, x”-‘f(gK
@ H,x) = (x-gn)n-lf(H,x-gn)
(9)
for any non-negative integer g. This last relation may also be derived directly from Theorem I or from Theorem II. Let bH denote the b-fold O-sum of a graph H with itself. It follows readily from definition (1) that f(bH, x) = f(H, x/b) for any positive integer b. If M denotes Vol. 316, No. 2, pp. 187-190, August Printed III Great Britain
1983
the graph with n vertices and no edges then 189
J. W Moon and S. D. Bedrosian f(M,z)
= 1 and kK 0 M = kK; thus it follows from (8) that f(kK,x)
=
(
I-;
>
n-i
for each positive integer k (see also (12; p. 54)). We may therefore introduce an additional parameter into relations (8) and (9) and express the extended versions as f(kK 0 bH,x) =f(kK,x)f
(
H,?
(10)
>
and ,
(11)
provided that bH is a subgraph of kK in relation (10). When k = b + 1 relation (10) is equivalent to a result derived by Kelmans (6; Eq. 13) in another way. Finally, we remark that the foregoing relations still hold even if the graph H has only m vertices where 1 < m < n; this follows from the observation that if H* denotes the graph obtained by adding n-m isolated vertices to H, then H* and H clearly have the same generic forms and the graphs kK 0 bH and gK 0 bH are the same as the graphs kK @ bH* and gK @ bH*. References (1) S. D. Bedrosian, “Generating formulas for the number of trees in a graph”, J. Franklin Inst., Vol. 221, pp. 313-326, 1964. (2) S. D. Bedrosian, “The Fibonacci numbers via trigonometric expressions”, J. Franklin Inst., Vol. 295, pp. 175-177, 1973. (3) S. D. Bedrosian, “Tree counting polynomials for labelled graphs. Part I : properties”, J. Franklin Inst., Vol. 312, pp. 417430, 1981. (4) D. M. Cvetkovic, M. Doob, and H. Sachs, “Spectra of Graphs”, Academic Press, New York, 1980. (5) W. Feller, “An Introduction to Probability Theory and Its Applications”, Vol. 1 (2nd edn), John Wiley, New York, 1957. (6) A. K. Kelmans “The number of trees in a graph, I”, Aut. Remote Control, Vol. 26, pp. 2118-2129, 1965. (7) A. K. Kelmans, “The number of trees in a graph, II”, Aut. Remote Control, Vol 27, pp. 233-241,1966. (8) A. K. Kelmans and V. M. Chelnokov, “A certain polynomial of a graph and graphs with an extremal number of trees”, J. Comb. Th., Vol B 16, pp. 197-214, 1974. approaches to network theory”, (9) Y. H. Ku and S. D. Bedrosian, “On topological J. Franklin Inst., Vol. 279, pp. 11-21, 1965. (10) J. W. Moon, “The second moment of the complexity of a graph”, Mathematika, Vol. 11, pp. 95-98,1964. (11) J. W. Moon, “Enumerating labelled trees”, in “Graph Theory and Theoretical Physics” (Edited by F. Harary), pp. 261-272, Academic Press, New York, 1967. Congress, Montreal, (12) J. W. Moon, “Counting labelled trees”, Canadian Mathematical 1970. (13) H. N. V. Temperley, “On the mutual cancellation of cluster integrals in Mayer’s fugacity series”, Proc. Phys. Sot., Vol. 83, pp. 3-16, 1964. Journal
190
of the Franklin
lnst~tute Ltd.
Pergamon Press
J. W. MOON
University of Alberta, Edmonton, Alberta, Canada T6G 2Gl and S. D.
BEDROSIAN
Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. ABSTRACT : Each graph may be associated with a certainfunction called its genericform. If one knows the generic forms of given graphs, then one can easily determine the number of spanning trees in graphs obtained from a complete multi-graph either (1) by adding, or (2) by deleting the edges of disjoint copies of the given graphs. Our objective here is to give a proof of a simple and useful relation between the generic forms of graphs that are complementary with respect to a complete multi-graph.
I. Introduction
All graphs considered here have vertices labelled 1, 2,. . .,n unless indicated otherwise; there may be several edges joining the same pair of vertices, but we shall ignore the presence of loops. For given graphs A and B, let A @ B denote the graph obtained by identifying the vertices (but not the edges) of A and B in the obvious way ; if B is a subgraph of A, then A 0 B denotes the complement of B with respect to A. For any positive integer k, let kK denote the complete multigraph (on n vertices) in which each pair of distinct vertices is joined by K edges. For any given graph G, let c(G) denote the number of different spanning trees in G, let e(G) denote the total number of edges in G, let l(G) denote the number of connected components of G, and let p(G) denote the product of the number of vertices in the components of G. The genericformf(G,x) of the graph G may be defined by the relation
f(G 4 = 1
P(F) (-
W4@
FGG
where the sum is over all spanning forests F of G; notice that F(G, x) is a polynomial in l/x of degree n - l(G). (We remark that x”- tf(G, -x) is the same as the function B:(G) defined by Kelmans (6) as a certain determinant ; the main step in showing the equivalence between the two definitions is implicit in (13; p. 7); see also (8; p. 203) and (4; p. 38).) Generic forms can be used to determine the number of spanning trees in certain graphs by appealing to the following results, valid for any graph G and any positive integer k, provided that H is a subgraph of kK in Theorem II. Theorem I. c(kK @ H) = k(kn)“-‘f(H, Theorem
OThe
II. c(kK 0 II) = k(kn)“-2f(II,
Franklin1nst1tute00164032/83$3.00+0.00
- kn). kn).
187
J. W Moon and S. D. Bedrosian The case k = 1 of Theorem II is implicit in Temperley (13); but it seems that Bedrosian (1) was the first to discuss both results in this generality (6). For the sake of completeness we shall give short proofs of these results in Section II ; our proofs are closely related to the proof given in Ref. (11) for the case k = 1 of Theorem II which is perhaps the case most frequently applied in practice. In order to exploit Theorems I and II one needs, of course, to know the generic forms of the graphs H that arise. Formulas for the generic forms of various graphs may be found, for example, in (7), (12; Section 6.4) and (4; Section 7.6). There is a simple relation between the generic forms of complementary graphs, so that if one knows the generic form of a graph H then one can easily determine the generic form of the complementary graph kK @ H. This relation may be formulated as follows : Main result. If H is a subgraph of kK, then xn-‘f(kK
0 H,x) = (x-kn)“-‘f(H,kn-x).
(4
Relation (2) was discovered by Bedrosian and illustrated in (1); further applications were given in (2,3,9). Our primary objective here is to give a short proof of this relation in Section III that is based on Theorems I and II. An alternative approach to a relation equivalent to (2), based on determinants, is given in (6; pp. 2125-2126); see also (4; p. 58).
II. Proof of Theorems For any forest F and any graph G, let t(F, G) denote the number such that F @ J is a tree. It is known (l&12) that
of forests J in G
t(F, K) = p(F)n”-’ -@). Since the forests J must have n- 1 -e(F)
edges, it follows that
t(F,kK) = k”-‘-“‘F’t(F,K)
= kp(F)(kn)“-2p”(F)
(3)
for any positive integer k. (These results hold even if F has no edges.) We now count the number of spanning trees in the graph kK OH. It is not difficult to see that for each spanning forest F in H, there are t(F, kK) spanning trees T in kK @ H such that T n H = F. It follows, therefore, that c(kK @ H) = 1
t(F, kK) = k(kn)“p2 c
=
k(kn)“-2f(H,
p(F)(kn)-“‘F’
FEH
FEH
- kn)
as required, by (3) and (1). To count the spanning trees in kK 0 H, we must exclude from the set of all spanning trees in kK those which contain any edges of H. It is not difficult to see that for each spanning forest F in H, there are t(F, kK) spanning trees Tin kK such that F 5 T n H. It follows, therefore, by the method of inclusion and exclusion (5; p. 96) that c(kK @ H), the number of spanning trees T in kK containing no edges of H, is 188
Journal of the Franklin Institute Press Ltd.
rergamon
On Generic Forms of Complementary
Graphs
given by the formula c(kK @ H) = =
1 (- l)“%(F, kK) = k(kn)“-’ FEH k(kn)“-‘f(H:
1
p(F)( - kn)-“cF)
FEH
kn),
as required.
III. Proof of Main Result Consider
the graph G=gK@H=(g+k)K@{kKOH}
(4)
when g is an arbitrary positive integer. If we apply Theorems second expressions for G, we find that c(G) = s(s$
- 2f(K
I and II to the first and
- gn)
(5)
and c(G) = (g + k) {(g + k)njnP2f(kK
0 H, (g + k)n).
(6)
Now consider the functions z”- ’f(H, -z) and (z + kn)“- ‘f(kK 0 H, z + kn), for fixed H, n, and k. Both functions are polynomials in z of degree n - 1. It follows from (5) and (6) that these polynomials are numerically equal when z = gn for g = 1,2,. . . ; hence, they must be identically equal for all z, i.e. z”-‘f(H,
-z)
= (z+kn)“-‘f(kKOH,z+kn).
If we let x = z + kn, this can be rewritten xn-lf(kK This completes
(7)
as
0 H,x) = (x-kn)“-‘f(H,
kn-x).
(8)
the proof of the main result.
IV. Concluding Remarks If we apply relation
(7) twice to the graph G defined in (4), we find that
(z-gn)“-‘f(ghOH,gn-z)=(z+kn)“-‘f(kKOH,kn+z)=z”-‘f(H,-z); hence, x”-‘f(gK
@ H,x) = (x-gn)n-lf(H,x-gn)
(9)
for any non-negative integer g. This last relation may also be derived directly from Theorem I or from Theorem II. Let bH denote the b-fold O-sum of a graph H with itself. It follows readily from definition (1) that f(bH, x) = f(H, x/b) for any positive integer b. If M denotes Vol. 316, No. 2, pp. 187-190, August Printed III Great Britain
1983
the graph with n vertices and no edges then 189
J. W Moon and S. D. Bedrosian f(M,z)
= 1 and kK 0 M = kK; thus it follows from (8) that f(kK,x)
=
(
I-;
>
n-i
for each positive integer k (see also (12; p. 54)). We may therefore introduce an additional parameter into relations (8) and (9) and express the extended versions as f(kK 0 bH,x) =f(kK,x)f
(
H,?
(10)
>
and ,
(11)
provided that bH is a subgraph of kK in relation (10). When k = b + 1 relation (10) is equivalent to a result derived by Kelmans (6; Eq. 13) in another way. Finally, we remark that the foregoing relations still hold even if the graph H has only m vertices where 1 < m < n; this follows from the observation that if H* denotes the graph obtained by adding n-m isolated vertices to H, then H* and H clearly have the same generic forms and the graphs kK 0 bH and gK 0 bH are the same as the graphs kK @ bH* and gK @ bH*. References (1) S. D. Bedrosian, “Generating formulas for the number of trees in a graph”, J. Franklin Inst., Vol. 221, pp. 313-326, 1964. (2) S. D. Bedrosian, “The Fibonacci numbers via trigonometric expressions”, J. Franklin Inst., Vol. 295, pp. 175-177, 1973. (3) S. D. Bedrosian, “Tree counting polynomials for labelled graphs. Part I : properties”, J. Franklin Inst., Vol. 312, pp. 417430, 1981. (4) D. M. Cvetkovic, M. Doob, and H. Sachs, “Spectra of Graphs”, Academic Press, New York, 1980. (5) W. Feller, “An Introduction to Probability Theory and Its Applications”, Vol. 1 (2nd edn), John Wiley, New York, 1957. (6) A. K. Kelmans “The number of trees in a graph, I”, Aut. Remote Control, Vol. 26, pp. 2118-2129, 1965. (7) A. K. Kelmans, “The number of trees in a graph, II”, Aut. Remote Control, Vol 27, pp. 233-241,1966. (8) A. K. Kelmans and V. M. Chelnokov, “A certain polynomial of a graph and graphs with an extremal number of trees”, J. Comb. Th., Vol B 16, pp. 197-214, 1974. approaches to network theory”, (9) Y. H. Ku and S. D. Bedrosian, “On topological J. Franklin Inst., Vol. 279, pp. 11-21, 1965. (10) J. W. Moon, “The second moment of the complexity of a graph”, Mathematika, Vol. 11, pp. 95-98,1964. (11) J. W. Moon, “Enumerating labelled trees”, in “Graph Theory and Theoretical Physics” (Edited by F. Harary), pp. 261-272, Academic Press, New York, 1967. Congress, Montreal, (12) J. W. Moon, “Counting labelled trees”, Canadian Mathematical 1970. (13) H. N. V. Temperley, “On the mutual cancellation of cluster integrals in Mayer’s fugacity series”, Proc. Phys. Sot., Vol. 83, pp. 3-16, 1964. Journal
190
of the Franklin
lnst~tute Ltd.
Pergamon Press