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<ul><li><p>Matchings and Walks in Graphs </p><p>C. D. Godsil INSTITUT FUR MATHEMATIK UND ANGEWANDTE GEOMETRIE </p><p>MONTANUNI VERSITAT LEOBEN, AUSTRIA </p><p>ABSTRACT </p><p>The matching polynomial a(G, x ) of a graph G is a form of the generating function for the number of sets of k independent edges of G. In this paper we show that if G is a graph with vertex u then there is a tree T with vertex w such that </p><p>This result has a number of consequences. Here we use it to prove that a(G \ Y, 1 / x ) /xa(G, 1 / x ) is the generating function for a certain class of walks in G. As an application of these results we then establish some new properties of a(G. x ) . </p><p>1. INTRODUCTION </p><p>Let G be a graph with n vertices. A k-matching in G is a set of k edges, no two of which have a vertex in common. If p(G, k ) denotes the number of k- matchings in G then the matchingpolynornial of G is </p><p>li 1 a( G, X ) = C (- 1 ) k p ( G, k ) ~ " - ~ ~ , </p><p>k = 0 </p><p>where we assume p(G, 0) = 1. If G has adjacency matrix A then the characteristic polynomial @( G, x) of G is det(x1- A ) . </p><p>The key result of this paper is that, given a graph G with a vertex Y, there is a tree T with a vertex w such that </p><p>Journal of Graph Theory, Vol. 5 ( 1 981 ) 285-297 0 1 981 by John Wiley &Sons, Inc. CCCO364-9024/81/030285-13$01.30 </p></li><li><p>286 JOURNAL OF GRAPH THEORY </p><p>The characteristic polynomial of a forest coincides with its matching polynomial (see [4]). Consequently many properties of a(G, x) can be established using ( 1), together with known results about the characteristic polynomials of graphs. For a detailed discussion along these lines, see [2]. </p><p>A second aspect of (1) is that the right-hand side of it can be viewed as a form of the generating hnction for the closed walks in T which start at IY. This observation enables us to prove that </p><p>x-' a( G \u,x-' ) a( G , x-' </p><p>is the generating function for a certain class of closed walks in G which start at u. This somewhat unexpected relation between walks and matchings does not appear to have been previously recognized. </p><p>We are able to use the above results to obtain further information about a(G, x). For example, we show that if G has n vertices then a(G, x) (and hence the number of 1-factors of G) is determined by the collection of subgraphs of G with [n/2] 4- 1 vertices. </p><p>2. THE MATCHING POLYNOMIAL AND T(G, v ) </p><p>We begin by presenting two preliminary results which can be used to simplify the task of computing the matching polynomial of a graph. Their proofs may be found in [4] and [ 5 ] . </p><p>Lemma 2.1. Let G be a graph with a vertex u. Let H = G \ zi and let w,, . . . wh be the vertices in G adjacent to u. Then </p><p>Lemma 2.2. Suppose G is the disjoint union of the graphs K and L. Then </p><p>a ( G , x ) = a(K , x)a(L, x). </p><p>Definitions 2.3. </p><p>(a) A walk of length n in the graph G is a sequence </p><p>of vertices from G such that pi is adjacent to Pi+, for i = 0,. . . , n - 1. If Po = PI, then p is a closed walk. We regard a single element as a closed walk </p></li><li><p>MATCHING AND WALKS 287 </p><p>with length zero. If no vertex is visited twice by 8 (i.e., Pi = Pj implies i = j ) then we call 8 apath. We use 8, to denote the walk </p><p>( P o , . . . 9 Pm), </p><p>where m I n. Thus if 8 has length n , 8 = 8,7. (b) Let G be a graph with a vertex v . We define T(G, v ) to be the graph </p><p>which has as vertices the paths in G which start at v and where two such paths are adjacent if one is a maximal proper subpath of the other. (Thus if 8 is a path of length n in G which starts at v then the only path starting at v with length less than n which is adjacent to 8 is 8,1-, .) The single element path ( v ) in T( G, v ) will always be labeled v . We note that T( G, v ) depends only on the component of G containing v . </p><p>Our next result gives some important properties of T(G, v) . </p><p>Lemma 2.4. Let G be a graph with a vertex v. Then </p><p>(a) T(G, v ) is a tree. (b) If G is a tree, T(G, v ) G G. (c) If wI , . . . , wk are the vertices in G adjacent to v then T(G,u) can be </p><p>obtained from the forest </p><p>by adjoining a vertex v and defining it to be adjacent to each of the vertices w;(l l il k ) . </p><p>ProoJ We prove (b) and (c) first. If G is a tree then the map taking each vertex w in V(G) onto the unique path joining v to w provides the required isomorphism from G to T(G, v) . </p><p>The last statement follows directly from the definition of T(G, v ) . We leave the details to the reader. </p><p>The proof that T( G, v ) is a tree proceeds by induction on n , the number of vertices in G. When n 5 2 the claim follows from (b). If n > 2 we assume inductively that if w E G\ZJ then T(G\v, w) is a tree. It follows then from (c) that T(G, v ) is itself a tree.1 </p><p>The next result provides our justification for introducing the tree T( G, v) . </p><p>Theorem 2.5. Let v be a vertex in the graph G. Then </p><p>and a( G, x ) divides a( T( G, v ) , x ) . </p></li><li><p>288 JOURNAL OF GRAPH THEORY </p><p>Froo$ Suppose G has n vertices. If G is a tree the result follows by Lemma 2.4(b). Hence the theorem is true if n i 2. If n > 2 we assume inductively that the theorem is true for all graphs with less than n vertices. Set H = G\v and let wI , . . . , wh be the vertices in G adjacent to v. Then </p><p>(by 2.1).1 </p><p>The matching polynomial of a tree coincides with its characteristic polynomial (see [4]). Hence the zeros of the matching polynomial of a tree are real. From Theorem 2.5 we conclude then that the zeros of the matching polynomial of any graph are real. For a more detailed discussion of this and some related matters, see [2]. </p><p>3. TREELIKE W A L K S </p><p>In this section we establish a relation between the matching polynomial and a class of walks in G. </p><p>Definitions 3.1. A walk 6 of length n 1 2 in G will be said to repeat about i (0 < i < n ) if pi-, = pi+,. In this case we can obtain a new walk, y say, of length n - 2 from 6 by setting </p><p>We will say 2 is obtained from 6 by reduction at i. </p><p>If 6 does not repeat about any i then we call 6 an irreducible walk. Since the process of reduction always gives rise to a walk which is shorter than the original walk, it follows that we can always obtain an irreducible walk from a given walk in a finite number of steps. In particular, we can always obtain an irreducible walk from 6 by a finite number of reductions, each of which is carried out at the lowest possible value of i. The resulting walk will be denoted by $. </p></li><li><p>MATCHING AND WALKS 289 </p><p>Our next result shows that any two irreducible walks obtained from a given walk are identical. </p><p>Lemma 3.2. Suppose (y is obtained from the walk 1 by a sequence of reductions. Then = $. </p><p>Proof. It will suffice to consider the case where cy is derived from by a single reduction, because the general case follows from this by a trivial induction argument. </p><p>Suppose @ has length n. If n I 2 there is nothing to prove. Accordingly we assume n > 2 and that the result holds for all walks with length less than n. Let the walk a be obtained from e by reduction at i and let j be the least positive integer such that e is reducible a t j . Clearly j I i. </p><p>If j = i, then y = ,@ by definition. Consequently we assume j < i. If j < i - 1 then y repeats aboutj. Suppose that by reducing both y and a t j we obtain walks y and !, respectively. Then y can be derived from ! by reduction at i - 2 and so, by our induction hypothesis, </p><p>Since i is the least index about which ,@ repeats we also have i= $. Therefore 6 = f i as claimed. </p><p>repeats about i - 1 and i. Thus we must have </p><p>Suppose now that j = i - 1. Then </p><p>for suitable distinct vertices u, v in G. It follows that the walk obtained from ,@ by reduction a t j = i - 1 is just y. As i - 1 is the least index about which e repeats, it follows that $ = 5 . B Definition 3.3. If ,@ is a walk in G with length n such that $,, is a path for each m, (0 I m I n) , then we call .@ a treelike walk. </p><p>We note that if ,@ is both closed and treelike, then ,@ is just a single vertex. However if is a walk in G and f i is a single vertex it does not follow that is treelike. For example, if </p><p>is a walk in G then </p><p>basic information about treelike walks. </p><p>= (v), but ,@ is not treelike. The next result justifies our choice of notation, as well as providing some </p><p>Lemma 3.4. The following statements about a graph G are equivalent: </p></li><li><p>290 JOURNAL OF GRAPH THEORY </p><p>(a) G is a forest (b) each walk in G is treelike (c) each closed walk in G is treelike. </p><p>h o $ Clearly (b) implies (c). I f (a) is false then G contains a circuit vo, 4,. . . , v,, where v; is adjacent to vi+l (0 l i I n) and v,, is adjacent to vo. The walk defined by setting </p><p>vi, O I i l n </p><p>vo, i = n - 1 Bi = </p><p>is then an irreducible closed walk in G which is not a path. Consequently (c) implies (a). </p><p>It remains to show that (a) implies (b). Let f i be a walk of length n in the forest G and let B denote the subgraph of G induced by the vertices Bi(O I i I n). Since G is a forest and B is connected, B is a tree. I fB is a path then it is easy to show that 6 is a path. I fB is not a path it must have at least three endvertices. In this case it is not difficult to show that there is an index i, 0 < i < n, such that Pi is an endvertex in B. Consequently ,@ repeats about i. </p><p>Therefore if B is not a path, ,@ is reducible. We conclude that if B is irreducible then B is a path. Hence 8 is always a path, and so each walk in G is treelike.1 </p><p>In view of the proof of 3.4 it is worth remarking that ifg is a treelike walk of length n in the graph G and G is not a tree, then the subgraph induced by the vertices Pi, 0 5 i 5 n, need not be a tree. For example, if </p><p>is a walk op G it is a treelike walk, even though the subgraph induced by the vertices in contains a cycle. </p><p>Theorem 3;s. Let G be a graph with a vertex v . Then there is a bijection between the set of walks in T(G, v ) starting at v and the set of treelike walks in G starting at v. This bijection maps closed walks onto closed walks and preserves walk length. </p><p>Prooj: Let sr be a walk of length n in T(G, v ) starting at v . We associate to ? a sequence =An) of vertices in G by defining pi, 0 5 i I n, to be the endvertex of the path in G corresponding to T,. We claim thatfis the bijection we need. </p><p>We show first that 6 is actually a walk of length n in G. If i is an integer, 0 5 i < 1 2 . one of the paths in G corresponding to the vertices T, and q+l is a </p></li><li><p>MATCHING AND WALKS 291 </p><p>maximal proper subpath of the other. Consequently the endvertices of these paths are adjacent, which implies that pi and are adjacent. Therefore p is a walk of length n in G. </p><p>We note that if, for some i, q = v then pi = v. Hence ,@ =A?) starts at v and is closed if? is. Thusfsatisfies the second claim in the statement of the theorem. </p><p>We prove now that f is an injection. Suppose ,K and $ are distinct walks in T(G, v ) which start at v. Let @ =A?), y = A$). We assume that @ = 1 and show in consequence that ? = 9, a contradiction. </p><p>Let i be the least non-negative integer such that ri = k. This integer exists, since ? # $. Also, as no = rt0 = v , i 1 1. By our choice of i, ri-l = 16;.-, , and since 8 = y, pi = yi. Hence, Ti and $[ are two walks of length i in G which start at v, have a common maximal proper subwalk, and the same endver- tices. Therefore they are equal and so it follows that ? = $. Thus, f is an injection. </p><p>is a walk in T(G, v ) starting at v and @ =AT) then@ =A@). Suppose and F ~ + ~ have the same endvertex. As these paths are both adjacent to Fi, it follows that these are equal. On the other hand, if repeats about i then so must p. Consequently ? repeats about i if and only if @ does. </p><p>Suppose then that ? repeats about i. Then 8 repeats about i and if$, y are derived from ?, @, respectively, by reduction at i, then y =A$). By induction on the length of? we conclude thatAn) can be derived from 6 by a series of reductions. By the previous paragraphA@) is irreducible and so, by Lemma 3.2,Ag) = 6. </p><p>We can now prove that @ is a treelike walk in G. Since T(G, v ) is a tree, ? is treelike by Lemma 3.4. Hence, is a path. To provet is treelike we must demonstrate that i ) =An) is a path. Let $ = @. </p><p>is a maximal proper subpath of $;, 1 i i 5 n. If is also a maximal proper subpath of 9, then = which is impossible since $ is a path in T(G, v ) . As $; and are adjacent we conclude that must be a maximal proper subpath of Using a simple induction argument on the length of $=@we find that,given i a n d j such that 0 5 i < j l n, $; is a proper subpath of &. Consequently the paths in G corresponding to 9; and % have distinct endvertices and so it follows that the vertices in @ are all distinct. Thus, 6 is a path. </p><p>We have shown, then. thatfis an injection from the set of paths in T(G, u ) starting at t i into the set of treelike walks in G starting at v. We complete the proof by showing that it is a surjection. </p><p>k t fI be a treelike walk of length n 1 1 in G which starts at u. Let y = p,,-l. We claim that one of 7, ,@ is a maximal proper subpath of the other. </p><p>Suppose that the length o f f is m. We define the sequence 6 of length ?n + 1 by setting </p><p>Our next step is to show that if repeats about i. Then </p><p>Obviously Po is a maximal proper subpath of 9,. Suppose </p></li><li><p>292 JOURNAL OF GRAPH THEORY </p><p>We see that _s can be obtained by a series of reductions from 6 and so is a treelike walk in G starting at v . We also have _s = E . </p><p>= ,@ and _r are both paths. If _s itself is a path then 5 = is a maximal proper subpath of $. If! is not a path then, since it is a treelike walk, it must repeat about some index i. Since _r is &reducible, it therefore repeats about m, but no other vertex. Consequently _s = a,,--, is a maximal proper subpath of j . </p><p>If] is a treelike walk of length n in G starting at v we therefore define a sequence </p><p>AS fi is treelike, = $ and so </p><p>of paths in G by setting </p><p>?Ti = g j (0 I i I n ) . This sequence has the property that rr, = v and that for each i, 0 5 i < y1 - 1, rj is a maximal proper subpath of T,,~, or vice versa. Accordingly we may regard p as a walk in T(G, v ) which starts at v . </p><p>Since $I and 6, have the same end-vertices we also find thatfl?) = e . Hence we have now shown thatfis a surjection from the walks in T(G, v ) starting at v onto the set of treelike walks in G starting at v . This completes the pro0f.D </p><p>Our next result provides a relation between the matchings and treelike walks in G. </p><p>Theorem 3.6. Let G be a graph with vertex v. Then </p><p>which start at u is (a) the generating function for the number of closed treelike walks in G </p><p>(b) the generating function for the number of closed treelike walks in G is </p><p>where a(H, y ) = (d/dy)cr(H, y ) , for a given graph H. </p><p>ProoJ: As already noted, the matching and characteristic polynomials of a tree are equal and so, by Lemma 2.1 of [3] (for example), </p></li><li><p>MATCHING AND WALKS 293 </p><p>is the generating function for the closed walks in T( G, u ) which start at u. Thus (a) follows at once from Theorems 2.5 and 3.5. The generating function for all closed tree-like walks in G is the sum...</p></li></ul>