Czechoslovak Mathematical Journal

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1 Czechoslovak Mathematcal Journal Mroslav Fedler Algebrac connectvty of graphs Czechoslovak Mathematcal Journal, Vol. 23 (973), No. 2, Persstent URL: Terms of use: Insttute of Mathematcs AS CR, 973 Insttute of Mathematcs of the Academy of Scences of the Czech Republc provdes access to dgtzed documents strctly for personal use. Each copy of any part of ths document must contan these Terms of use. Ths paper has been dgtzed, optmzed for electronc delvery and stamped wth dgtal sgnature wthn the project DML-CZ: The Czech Dgtal Mathematcs Lbrary

2 Czechoslovak Mathematcal Journal, 23 (98) 973, Praha ALGEBRAIC CONNECTIVITY OF GRAPHS*) MIROSLAV FIEDLER, Praha (Receved Aprl 4, 972). INTRODUCTION Let G = (F, E) be a non-drected fnte graph wthout loops and multple edges. Havng chosen a fxed orderng w^, W2,..., w of the set V, we can form a square n-rowed matrx A(G) whose off-dagonal entres are а^-^ = aj,î = f (w^-, WJ^G E and ük = 0 otherwse and whose dagonal entres ац are equal to the valences of the vertces W;. Ths matrx Ä(G), whch s frequently used to enumerate the spannng trees of the graph G, s symmetrc, sngular (all the row sums are zero) and postve semdefnte {Ä{G) = UU^ where U s the (0,,-) vertex-edge adjacency matrx of arbtrarly drected graph G). Let n ^2 and 0 = Я^ ^ Я2 = ^(^) ^ >^з ^... g /l be the egenvalues of the matrx A(G). From the Perron-Frobenus theorem appled to the matrx (n - ) / Ä{G) t follows that a{g) s zero f and only f the graph G s not connected. We shall call the second smallest egenvalue a(g) of the matrx Ä(G) algebrac connectvty of the graph G. It s the purpose of ths paper to fnd ts relaton to the usual vertex and edge connectvtes. We recall that many authors, e.g. A. J. HOFFMAN, M. DOOB, D. K. RAY-CHAUD- HUR, J. J. SEIDEL have characterzed graphs by means of the spectra of the (O, ) and (0,, ) adjacency matrces. 2. NOTATION AND CONVENTIONS The notaton ntroduced above s used throughout the present paper. All matrces and vectors are consdered real. The transpose of a matrx M s denoted by M^, the dentty matrx by /, the vector (,,..., )^ by e, the unversal matrx ее''' by J, the cardnalty of a set 5 by 5. For our purpose t s convenent to denote by W the set of all column vector^ x such that x'^x =, x^e = 0. Any square matrx M wth all zero row sums has an 298 *) Presented at the Graph Theory Meetng n Zlata Idka, May 97.

3 egenvalue О and a correspondng egenvector s e. If M s postve semdefnte then the second smallest egenvalue s equal to mn x^mx by the well known Courant's theorem. We use that prncple tactly. ^^^ Further, let us menton two common concepts. Edge connectvty of the graph G,.e. the mnmal number of edges whose removal would result n losng connectvty of the graph G, s denoted by e{g). Vertex connectvty whch s defned analogously (vertces together wth adjacent edges are removed) s denoted by v{g). It s convenent to put v(g) = n for the complete graph G. Let Gl = (Fl, ), G2 = (F2, E2). By G x G2 we denote the graph {V^ x F2, E) such that ((wj, U2), (I'l, ^2)) e f and only f ether u^ = V and («2,^2) ^ ^2 or (w, ^) G Ej and и2 = V2' Let R = (r,y), S be matrces. Then by ^ x 5 the parttoned matrx {rjs) s denoted. 3. PROPERTIES OF a{g) 3.. // Gl, G2 are edge-dsjont graphs wth the same set of vertces then a(g) + + a{g2) й a{g, u G2). Proof. We have A{G^ u G2) = ^(G) + A{G2). Thus a{g u G2) = = mn (x"^ ^(G) X 4- x"^ ^(G2) x) ^ mn x'^ A{G) x + mn x^ ^(G2) x = Ö(GI) + + a(g2) Corollary. The functon a{g) s non-decreasng for graphs wth the same set of vertces,.e. a(g^) ^ a(g2) f G^ Ç G2 {and Gj, G2 have the same set of vertces) Let G be a graph, let G^ arse from G by removng к vertces from G and all adjacent edges. Then () a(g) ^ a{g) - k. Proof. Let G have n vertces and let G arse from G by removng one vertex, say u. Defne a new graph G' by completng n G all mssng edges from м. Then 4-)=(?''^\-',)- Let V be an egenvector of ^(G) correspondng to the egenvalue Ö(GI). Snce 4G')(o) = WGx)+l](o). a(g) + s an egenvalue of A(G') dfferent from zero,.e. ß(G') g a(g)

4 By 3.2, a(g) ^ a(g') whch mples () for Ä: =. The general case follows by nducton We have a{g x G2) = mn (a(g), «(Оз)). Proof. Let Gl = (Fl, ), G2 = (Kj, 2)- Order the set V^ x V2 lexcographcally. Then ^(G X G2) = ^(G) x /2 + / x ^(G2), Ij beng the Fy -rowed dentty matrx. By a well known result from the matrx theory [] all egenvalues of ^(G x X G2) are of the form ц + v where /x, v resp. are egenvalues of ^(G), ^(G2) respectvely. Hence the second smallest egenvalue of ^(G x G2) s ether a(g) + 0 or 0 + a(g2) Let G = (F, ), Vj be the valency of the j-th vertex. Then a{g) S [nl{n - )] mn V g 2 /(n - ). Proof. Snce n mn V g 2]^» = ^^? the second nequalty s true. The frst s an mmedate consequence of the followng lemma. Lemma. Let M = (шц^ be a symmetrc postve semdefnte n by n matrx such that Me = 0. Then the second smallest egenvalue À2 of M satsfes (2) ^2 S ["/('î "- )] nn тц. Proof. Observe that (3) À2 mn x^mx. Let us show that the matrx M = M - À2{I - п-ч) s also postve semdefnte. Let у be any vector n. Then у can be wrtten n the form у = c^e -{ C2X where. Snce Йе = 0, t follows that ут^му = clx'^mx = cl{x'^mx ~ Д2) è 0 by (3). Thus the mnmum dagonal entry of Й s nonnegatve: and (2) s proved. mn тц ^2( и~^) ^ О Remark. A matrx M = (m,/t) satsfyng condtons of the precedng lemma l\as also the property that the numbers ^Jm^ fulfl the polygonal nequalty,.e maxmJ/^ й^тг

5 Ths follows easly from the well known fact that M can be consdered as the Gram matrx of a system of n vectors u^,...,u wth sum zero: or Then we have for the lengths ^k = (U, Uk) = ujuk, X" = ^ l" ^ Z"/c for =,..., n, 2 max w,- ^ ]^ м^ and w = (wf, u^^^' = m\l^ yelds the result. If we apply (4) to the matrx M = M ^2( (l/n) J), we obtan 2 max (m - [(n - l)/n] Я,)'/^ ^ Яш,, - [(n - l)/n] Я,)»'^. In terms of the graph G, we obtan 2 max [m^, - (n - ) a{g)y^^ й E[nt;, - (n - ) Û(<^)]'^' For sake of completeness, we formulate the asserton 3.6. For the complete graph K wth n vertces a{k^ = n. In the followng theorem, we denote by b(g), for a graph G wth n vertces, the number b{g) = n - a{g) where G s the complementary graph to G The functon b{g) has followng propertes: b{g) s the maxmum egenvalue of Ä{G) or, equvalently (5) b{g) = max x^ A{G) x ; we have thus (6) a{g)ub{g), wth equalty f and only f G s a complete graph or a vod graph (.e. wthout edges); T b{g) = max b{g,) f Gj,..., G^ are all components of G; 30

6 y b{g^) й b{g2) f Gl ç G2,.e. V^ ç V2 and E^ ç 2 ^here =,2; G^ = (F^, E^), 4 b(g и G2) ub{g) + b(g2) - a(g n G2) where both graphs G^ and G2 are consdered to have the same set of vertces; 5 [nl{n - )] max v^g) й b{g) S 2 max v^g) where '(G) means valency of the -th vertex n G. Proof. If G s the complementary graph to G (wth n vertces so that G has also n vertces) then A{G) + A{G) = nl - J. Snce a(g) = mn x^ ^(G) x and x^{nl J) X = n for, we have max x^ A(G) X = n mn x^ A(G) x = n a(g) = b(g). Ths mples (6). Let equalty be attaned n (6). Then x^ A{G) x s constant on Ж Takng frst x = [n(n )]~/2 (je. e) where e,- has all coordnates zero except the f-th equal to one, we obtan that all the dagonal entres n A{G) are equal. Choosng then x = 2"^^^^^ I'^^^Cj, { Ф k), we obtan that also all off-dagonal entres of A{G) are equal, thus all equal ether to, or to zero. Ths proves, 2 follows easly from snce A(G) s the drect sum of A{G^) f G s not connected and Gl are components of G. To prove 3 and 4, we shall use (6). Ths mphes 3 mmedately whle 4 follows from 4GI U G2) = A{G,) + ^(G2) - A{G^ n G2). The rght nequalty n 5 follows from and the well known fact that the maxmum modulus of the egenvalues s less than or equal to any norm. The maxmum norm (.e. wth respect to the vector norm jc = max \x\) of A[G) (known to be max ^ \ak\) s 2 max V(G) whch yelds ths result. To prove the left nequalty n 5, к _ let us apply 3.5 to the complementary graph G. We obtan whch can be wrtten as a{g) S [пцп - )] mn V{G). 302 n - b{g) S [пцп - )] [w - - max vlg)].. Ths mples the nequahty and the proof s complete.

7 3.8. We have a(g) ^ 2 mn V{G) - n + 2. Proof. Follows mmedately from the rght nequahty n 5 of 3.7 used for the complementary graph G Let G wth n vertces contan an ndependent set of m vertces (.e. no two of them joned by an edge of G). Then a[g) ^ n m. Proof. If G contans an ndependent set of m vertces then G contans a complete subgraph K^, Snce b{k^) = m, we have by 3 of 3.7 that so that a{g) = n - b{g) ^ n - m. b{g) ^ m 3.0. // G /5 a graph wth n vertces whch s not complete then a(g) ^ n 2.. Proof follows mmedately from // Kp q denotes the complete bpartte graph the parts of whch contan p and q vertces then a[kpq) = mn (p, q). Proof. Follows from 2 of 3.7 appled to the complement of Kp^ Let G = (F, ), let V = V^ ^ V2 be a decomposton of V, let G ( =, 2) be the subgraph of G generated on F^. Then a{g) ^ mn {a{g,) + F^j, a{g2) + \V,\). Proof. Ths s just a symmetrc formulaton of RELATIONS BETWEEN a{g), e{g) AND v{g) 4.. Let G be a non-complete graph. Then a{g) ^ ^{G). Proof. Let G = (F, É) and let V^ be a vertex cut such that F2 = F- Fj ф 0. Snce the subgraph G2 generated by G on V2 s not connected, we have by 3.2 Ths mples the asserton. <G)u\Vr\. 303

8 4.2. We have v{g) ^ e{g). Proof. Ths well known nequalty s an easy consequence of the followng theorem [3]: Let w, w' be a par of vertces of G. Then there exst v(g) paths between w, w' n G, no two of them havng any vertces n common (except w, w') Let СI = 2[cos (тг/п) cos (2л;/п)], C2 = 2 cos (я/и) ( cos {njn)) and let q(g) be the maxmum vertex valency of the graph G. Then a{g) ^ 2 e{g) ( - cos (тг/п)), fl(g) ^ C e(g) - C2 q{g), t/e second bound beng better f and only f 2 e{g) > q{g), Proof. Consder the egenvalues (J ^ <Т2 ^... ^ c^ of the matrx S = (sj) = = / q^^g) Ä{G). Ths matrx s symmetrc and stochastc (.e. ts row sums are and the entres are nonnegatve so that a^ = ). Denote by ^ the "measure of rreducblty" of S, the number mn ^ s^^, where V = {, 2,..., n}. Then accordng 0-2 à 2( COS (л;/п)) л, - СГ2 ^ - 2( - ^) COS (тг/п) - {2f - ) cos (27г/п). We have (72 = - a(g)/^(g), /x = e(g)/^(g), thus «(G)/^(G) ^ 2( - cos (тг/п)) e(g)/g(g), a{g)lq{g) ^ - 2( - e{g)lq{g)) cos (тс/п) - (2e(G)/g(G) - ) cos (27г/п), whch mples the requred nequaltes. The last asserton s easy to verfy We have the followng values for some types of graphs. graph path crcut star. complete g. cube (m-dmensonal) a(g) 2(- cos (n/n)) 2(-- COS (2n/n)) n 2 e(g) 2 w- m v(g) 2 n- m bound of 4.3 2( - cos (n/n)) 4 sn^ (n/n) 2( - cos (n/n)) 2(n - ) sn^ (n/n) 2m sn^ (n/2m) 304

9 Remark. After havng fnshed ths paper the author was nformed that W. N. ANDERSON, Jr. and T. D. MORLEY had obtaned some of these results n the paper Egenvalues of the Laplacan of a graph, Unversty of Maryland Techncal Report TR-7-45, October 6, 97. References [] Mc Duffee: The Theory of Matrces. Sprnger, Berln 933. [2] M. Fedler: Bounds for egenvalues of doubly stochastc matrces. Lnear Algebra and Its Appl. 5 (972), [3] H. Whtney: Congruent graphs and the connectvty of graphs. Amer. J. Math. 54 (932,) Author's address: 5 67 Praha, Ztna 25, CSSR (Matematcky ustav CSAV v Praze). 305