Sysło, Maciej M.

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Adjacency matrix equations and related problems: Research notes

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Commentationes Mathematicae Universitatis CarolinaeMaciej M. SysłoAdjacency matrix equations and related problems: Research notesCommentationes Mathematicae Universitatis Carolinae, Vol. 24 (1983), No. 2, 211--222Persistent URL: http://dml.cz/dmlcz/106220Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1983Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTAT!ONES HATHEMATICAE UNIVERSITATIS CAROLINAE 24.2(1983) ADJACENCY MATRIX EQUATIONS AND RELATED PROBLEMSiRESEARCH NOTES MACIEJ M.SYStO Abstracts The purpose of this paper is to discuss some generalizations of the notion of regularity in graphs which are derived from matrix equations involving adjacency matri-ces. Key words and phrases: Adjacency matrix of a graph, de-gree sequence, r-degree sequence, matrix equation, regular and semi-regular graphs, spectral technique. Classification: 05C50 Let Ĉ-v (k2*l) denote the class of all graphs such that G e C ^ if there exists an integer constant r for which A /fl = « r1l holds, where A is the adjacency matrix of G and 'll deno-tes the vector of l's. Evidently, G e t̂,_ if and only if G is regular and (#* consists of regular and semi-regular biparti-te graphs. For k>3f we have (fry. * <5-2 i f k *s eve*- Q-1*3- ^-^ * m (̂ 1 if k is odd. In the class of directed graphs, only some partial solu-tions of the equation A H = r'll are known. Some other matrix equations are also studied. Let G • (V,E) be a graph and dp(v) » 2l_, d(u), where Pv is the set of neighbours of v; dp(v) is called a P-degree of v. Evidently, there exists an integer r such that d„(v) -» r for every vertex v of G if and only if G e (̂ 2. In the last 211 -sseoti m., we consider a problem e£ finding when m s«quenee •£ m integers feme a P-degre« ••quemo* of some graph em n Tertieee?* 1* Squatiom A k4 « r^ in the class of simple graphs Let G he a simple graph (1*«# f a symmetric graph with-out loops and multiple edges) and A denote its adjacency ma-trix. If 4 m (lflf...fl) and dr - (d-^d^*.. td^) denote respectively the Teotor •£ m ernes mad the degree sequ«nom of G then we have eriimmtly A 4 « dl . A graph G is .regular if *{#> )» r for OTery Tertex T of Gf that is if A1 m ri . A grilUfl is celled seiiirmgular of degrees p9 q if G is hipertite #imh Tertex has degree p «r qt and each edge oenmeets a Y«rt*s)» of degree p with m TOI tern of degree q. PZonka L6J «m& Emma Chmndrma tel haTO considered the fol-lowing generalisation of the regularity in graphs. Let where T T denotes the set ef mmigmbmurs of T. Then m graph G --» r~rmgular (PXonka) or HDfl regular (Emma Chandran) if the-re exists m positive integer r such that OLXT) • r for every Tertex T of G. They proTed the following theorem Theorem 1.1 ([61 and L8]). A graph G is P-reguiar of degree r if and only if every connected component of G is ei-ther lTr-»regular or semiregular of degrees pt qf where p-q -- r. O One can easily notice that G is P -regular of degree r if and only if A241 - P K t since A-A1I - AdL. - 212 The main purpose of this section is to characterise pie graphs which satisfy (1) A 11 * rl for some positive integer rf where k ^ 3 . Let Vfc (k£l) denote the class of all graph* which sa-tisfy (l)t Hence, Q ^ is the class of all regular graphs aad ^? consists of P-regular graphs. In terms of graphs, Q be-longs to t k̂ if the number of paths of length k outgoing fron a vertex v is the same for all vertices of G • Que oa* tart-ly check that C^2 ft ̂ 3 since &2f3 6 Q"2 a n d ^2,3 ̂  ^3* S#m* relations between the classes Q k > k^l can be obtained by sim-ple matrix calculations. Proposition 1.1. For every k£l, G e ^rs ^fc+i if «*-* only if G is regular. Proof. Let G be s-regullar, i.e., ki . s'f # therefore Ak4f . A**!! . A ^ s f - s A 1 ^ -..* - Bk1l end A*+l1 . « sk+1i . Otherwise, if G e ̂ k n < ^ k + 1 then there exist %w© positive integers r-, and r2 such that Ak 4! « r^t and A + 1f • - v2i . Hence, Ak+1/ll » A»Ak/fl « A* ri^ • r i A ^ a a d r 2 ^ * r o . r-jÂ  . Since A is an integer matrix, we obtain k$ « — i , 1 r2 where — is a positive integer. Therefore G £ Q-l* D By similar calculations one can prove the following pro-perties of £ k (k£l). Proposition 1.2. If G e Q^k r\ <^£ then G e (f j ^ . D Proposition 1.3. C ^ c (^ k^ for every k . 2f3f... and -£« 2,3,... . a Proposition 1.4. G e (̂  k n ^ if g ^ o n l y if G e Q, m, - 213-where » - OCD(k,/). Proof. If Ge Cj. ̂ n Cj-g then let us assume that £ ^ % and apply the Buclid algorithm to JL and k (- k^): 2 m m^L^ + k2# where k ^ k^, k, « itgkg + k,, where k^^ k2, kp - V i V i ' kp+l " WD(kti); From the f irst equality, i t follows that a Hukn+k- k 9 nuk-, m, k.. A*1 - A x x H « A Z .A x *4 - r ^ A 2H , where r, i s » K l K l k-, k« m, oonstant such t ha t A x 4 - r, 41 . Hence, A d\\ - r / r ^ and n kl r/rv is the integer where A£/H « r fl . Therefore G € (L. • Kl ^*2 Iterating this process we obtain GeQ., t G e ^ t**«» ** e eC^kp+l "^GCD(k,£)-Otherwise, if G^^(jcD(k ^ ) * then» *>y Proposition 1.3t G e^.^ and G € <^ , since there exist integers k' and >£' such that k « GCD(k,i ) • k# and i - GCD(k,i ) • if . O Corollary 1.1. ^ ^ Q# if «-dL only if C^k • " ^GCBU-i)- ° Notice that the corollary is not equivalent to say that k - GCD(k,£ ). Let Z m 6, and k * 2,3. The above propositions show that ^ l f 9-2^3 c 9-g aad 9 2° 9-3 " 9-1 • We know also that 9-3 +• ̂ 6» s i n c e otherwise <#2 c C contradicting the ob-servation that K2 ~ e C$. 2 ~ 9-3* *"** suggests therefore that maybe <#6 - 9 2 and» i n ««-*ralf <*2£ - 9h2, for i 2 2 , 214 -*wt±e* ttat if It Isolde, thenf by the corollary, ^ k • Q-i for every odd integer kf since ^ j cc 9- 2k * ̂ 2 * This conjecture was proved by Cvetkovio and Doob C U by using spectral technique 12]. Kotiee that if for a symmetric 0-1 ma'erix Af there exist integer constants k and r such that A* I » r1 then r is an eigenvalue of A and I is ac eigen-vector of A beloiflging to the eigenvalue r. Theorem 1.2. for k > l f (̂ ^ « Q-, if k i« oddf and ^ K * • ^ 2 if k Is even. O It would be interesting to prove this theorem without u-sing spectral technique, either by matrix calculations or by applying sorae graph-theoretic results. 2. Other matrix equations. Lam £3J has considered the following equation (2) Ak m ocl + fi J, where A is a Ofl~raatrixf I denotes the identity matrix, J de-notes the raatrix with all entries equal to 1 and kfoc f/3 are integers. He proved that if A satisfies (2) then it has con-stant row and column sums. Hence if A is the adjacency raatrix of a simple graph G and satisfies (2)fthem G is regular. The-refore, the equation (2) with a symmetric 'matrix A produces only trivial solutions of (1). Let d+ (v) - dr(v) + d(v) and d~ (v) =- dp(v) - d(v). A graph G is P ̂ -regular ( F "-regular) of degree r if d"t, (v)=-» r (d~r(v) « r, respectively) for every vertex v of G. PXon-ka 111 has proved that G is P+-regular if and only if G is - 215 -regular. However, for P""-regular graphs, only some partial cha-racterizations have been obtained f7j and degree sequences of such graphs have been studied in £43. Votioe that 6 is P"^-regular if there exists an integer r guoh that (3) (A2 + A)H - ri holds, and G is P""-regular if there exists an integer s such that (4) (A2 - A H - mi holds. One may again generalize P+-regular graphs and ask for graphs which satisfy the equation (5) (Ak + A*"1 + ... + A)1l - HI for some integer r. In terms of elements of graphs, G satis-fies (5) if and only if the number of paths of length not gre-ater than k outgoing from a vertex of G is the same for all Tertioee of G. We conjectured that only regular graphs satisfy (5) for every k£l, and it has been settled in the affirmative by B. McKay [53 using again spectral techn ique. , -*• Equation A 1 » r1 in the class of digraphs. Let D « • (T,I) be a digraph, where Ec Vx v. One may now ask for the •dilation of the equations, (1), (3)»(4) and (5) when A is an ad-jaeemoy matrix of a digraph, that is, when A is an arbitrary 0-1 matrix. Let 9) jj. denote the class of all digraphs such that D ^ S ) ^ if its adjacency matrix satisfies (1). There are many different - 216 -types of digraphs in 3 ^ and a complete characterisation of 2) k seems to be Tery difficult. B, McKay [51 found the complete solution of (1) in the class of etrongly connected digraphs. A digraph D is regular oyclicftlly r-partite if V can be deoomposed V « V-jUVg1^..* M . u Y r , where v . t n V j • 0t V ^ 0 (it 3 - lt2t...tr)t and the-re exist integers m, tnut... tm such that for eTery T C V . and 1 • lt2t...tr we have r m,t i f j as i+1 (mod r) I E O H T S X V,)l « i ± 3 *- 0, otherwise. Brery regular digraph is regular cyclically 1-partite and re-gular cyclically 2-partite digraphs are exactly semiregular bipartite. Theorem 3»1 (B. McKay). If D is strongly connected then D e Q ^ if and only if D is regular cyclically r-partite^ for some r| k. 2 Dn Fig. 1 Figure 1 shows a digraph D which is not strongly con-nected and A41 « (n-2)1| for eTery n>3» ETery strongly con-nected member of 2) k can be used for a similar construction. 217 "tŞ 'IV — ГJl/ ,\ 4. P-graphical sequences. It is one of the classical problems in the graph theory to ask when a sequence of n inte­gers d » (d-j ,cU,.. * p&J) is graphical, i.e., constitutes the sequence of vertex degrees of a graph, which is usually assum­ed to be from a restricted family of graphs. In a similar way, we say that dl/« (d^9d^,«.. ,dn) is P-graphlcal if there exists a simple graph G such that d/ is the sequence of P-degrees of G. The problem of finding when a se­quence of n integers is P-graphical geems to be very hard. We present here only some comments. First notice that if dL' is P-graphical then O TV ,where dl m (d,,cL,... ,dL) is the degree sequence of a graph having P-degree sequence dV-Therefore, if d ' is P-graphical then it has the follow­ing properties! (a) R "*?/} d^ is even, (b) R can be deoomposed into n squares of integers cL, o^,. • • t^n» (c) cH » ( d . p d g , . . . ,dn) is graphical. It is not difficult to find a non P-graphical sequence d which satisfies (a) - (c). For instance, let dL* * (4,2,2,2). In this case R « 10 and there is only one decomposition of 10 into 4 squares, namely 10 * 2 + 2 + l2 + l . The sequence (2,2,1,1) is the degree sequence of P, which however has the P-degree sequence (3,3,2,2), different from the sequence we started with. Notice that in the class of multigraphs with loops, the sequence (2,2,1,1) has a realization (consisting 218 of P3 and a loop) with the given P-degree sequence. Therefore we have to add the following requirement (d) the sequence & has a realization with F-degree sequence dl'» Hotice that for every even integer R there exists a F -graphical sequence di » (d, f d . 2 f . . . fd ) such that .2^ dj « R. It is sufficient to take n « R and &^ « 1 for every i » lf2f ...fn$ oH corresponds to the graph on n vertices and n/2 se-parate edges. One may ask now which even integers have con-nected realizations. For instance if R « 8 then we have R m I2 + I2 + I 2 + l2 + I 2 + I 2 + l2 + l2f R M 2 2 + 2 2 f R « 2 2 + I2 + I2 + I2 + l2f and only the first decomposition corresponds to a graph. The-refore, 8 has no connected realization. It can be easily proved that if dL* is P-graphical then the number of 1 s in d is even and they can be removed from dl', since dl' is F-graphical if and only if di* without an even number of 1 s is P-graphical. Elements of dL equal 2 may occur in two configurations shown in Fig. 2f which however cannot be recognized from a F-degree sequence. If G is assumed to be connected and has at least four vertices, then only 2(b) may occur, that isf e-very vertex of P-degree 2 is pendant and adjacent to a ver-tex w such that d(w) * 2 and d (w) *g 3* H4=0 Fig. 2 219 (b) We have been able to «ntnwirate all graphs with F-deg-ree sequence dl' * (d^,dg,... ,d^) satisfying d ^ 5 (i •» l»2f ...,n). For instance, if d̂ -fe 4 (i » lf2f...fn) then every component of a graph with P-degree sequence dl is one of the graphs: a simple path, a simple cycle, K, ., and P. with an edge attached to a non-pendant vertex of ?,. Let us return to the conditions (a) - (d). The first of them can be trivially checked. The second one, however, is not trivial at all. First, some integers, usually of small value, cannot be decomposed into a fixed number of squares of integers. For instance, no sequence of five integers with the sum equal to 18 Is P-graphical since 18 cannot be decom-posed into 5 squares of integers. There are several such re-sults in the number theory (see for instance [9])t which can be used to eliminate some sequences of integers as non P-graphical. Secondly, if the sum R is decomposable into squa-res then the number of decompositions can be very large. In this case we have to find whether among the decompositions there exists at least one which gives a graphical sequence Compatible with the given P-degree sequence. However, neit-her in the number theory nor in the theory of graphs there exists a result dealing simultaneously with the decompos±ti~ on of an even integer into squares of integers and graphical realizations of the sequence of these integers. We have not succeeded (even restricting our attention to trees) in finding for P-degree sequences counterparts of d-invariant operations which can be applied to degree sequen-ces or to graphs, for instance, the removal of a vertex of maximum degree (as in the theorem of Havel and Hakimi) - 220 -and switching two mutually non-adjacent edges (see for ins-tance [101). Figure 3 shows that non-isomorphic trees can have the same P-degree sequence. i H».IM« i  J • 4 *• • * • •» » •'•On •• 4 "' • • * * * • i Pig. 3. It is known that if dt» (d-^dg,... ,0^) f d ^ l (i - 1»2, ...,n) and ,2 d, a 2(n-l) then every realization of dl in the class of connected simple graphs is a tree. It does not hold however for P-degree sequences. For instance, the sequ-ence of five 4's corresponds to K.. . and to C--. We conjecture that if a P-degree sequence dl has a tree realization then each such a realization of dl has the same number of pendant vertices. It is easy to check that this is not true for dis-connected realizations of P-degree sequences. R e f e r e n c e s [1] D.M. CVETKOVTC and M. DOOB: Personal Communication, May 1980. [2] D.M. CVETKOVTC, M. DOOB, and H. SACHS: Spectra of Graphs, VEB, Berlin 1980. [3] C.W.H. LAM: On some solutions of Ak - dl + X J , J. Com-binatorial Theory 23A(1977), 140-147. [4] Z. MAJCHBR: On sequences representable by P-regular graphs, in: M. Borowiecki, J.W. Kennedy, and M.M. Sys^o (eds.), Graphs in the Memory of Kc and K~ ^, Springer-Verlag, Berlin 1983. 221 -[5] B.MoKATs Personal Coawonications, September 1980 and July 1981. [6J J. ¥%OSKkt 0a r-regular graphs, Colloquiua Math. 46(1982), 131-134. [73 J. PJ-OHKAs On some regularities of graphs I, Sasopis pro pest, matematiky 107(1982), 231-240. [83 S. RAMI CHAHDRAHi Nearly regular graphs and their reconst-ruction, Graph Theory Hewsletter 8(1978), Ho. 3. [93 M. SIBRPIHSKI: Huaber Theory (in Polish), Vol. 2, 1959. [10 M.M. SYSXO: On tree and unioycllc realiiations of degree sequences, Demonstratlo Mathematlca 15(1982), la press. Institute of Computer Science, University of Wroclaw, ul. Przemyckiego 20, 51151 Wroclaw, Poland (Oblátům 2.2. 1983) - 222 -

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Adjacency matrix equations and related problems: Research notes

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