An Introduction to Relevant Graph Theory and Matrix Theory

Transcription

1 Chapter 0 A Itroducto to Relevat Graph heory ad Matrx heory hs boo s cocered wth the calculato of the umber of spag trees of a multgraph usg algebrac ad aalytc techques. We also clude several results o optmzg the umber of spag trees amog all multgraphs a class,.e., those havg a specfed umber of odes,, ad edges, e, deoted Ω (, e). he problem has some practcal use etwor relablty theory. Some of the materal ths boo has appeared elsewhere dvdual publcatos ad has bee collected here for the purpose of exposto. I preparato, we frst collect some relevat graph theoretcal ad matrx theoretcal results. 0. Graph heory hough we assume that the reader of ths wor s well versed Graph heory, ths secto we provde the prmary graph theoretc deftos ad operatos that are used the body of the succeedg chapters. For ay other termology ad otato ot provded here we refer the reader to Chartrad, Lesa ad Zhag [Chartrad, 0]. A multgraph s a par M = ( V, m), where m s a oegatve tegervalued fucto defed o the collecto of all two-elemet subsets of V, deoted V (). I the case there are several multgraphs uder

2 Spag ree Results for Graphs ad Multgraphs cosderato we use the otato ( ) M = ( V M, m M ). he elemets of V are called the odes of the multgraph. We assume that V s a fte set ad = V s the order of the multgraph,.e. the order s the umber of ( ) m u, v 0 ; odes. A multedge s a elemet { u, v} V() such that { } m( { u, v }) s called the multplcty of the multedge. If for each {, } m( { u, v} ) { 0,}, the M s a graph, ad m ({ } ) u v, = E s called the edge set. I ths case, we use G stead of M ad employ the alterate otato ( V, E) G =. Remar 0. he set of multgraphs cludes the set of graphs. hus ay result that s stated for multgraphs also holds for graphs. O the other had, results that are stated for graphs are oly applcable to graphs ad do ot hold for multgraphs. A multgraph has a geometrc represetato whch each elemet (ode) of V s depcted by a pot, ad two pots u ad v are oed by ({, }) m u v curves. Fgure 0. shows the geometrc represetato of a multgraph M ad a graph G. It s tradtoal to refer to each pot the represetato as a ode ad the collecto of all such pots as V. Also we refer to each curve the represetato as a edge, ad the collecto of all such curves as E. I the case that ({ }) m u, v we represet ay sgle edge betwee the odes u ad v by uv. he umber of edges, deoted by e, s the sze of the multgraph,.e. e E m( { u, v} ) = =. { u, v} V ( )

3 A Itroducto to Relevat Graph heory ad Matrx heory 3 We deote the class of all multgraphs of order ad sze e by Ω ( e) I Fgure 0. (5,0) M Ω ad G Ω ( 5,7),. M G Fgure 0.: A multgraph M ad a graph G. Note that a multgraph may have several geometrc represetatos that loo dfferet. We say that the multgraphs M ( V, m ) (, ) = ad M = V m are somorphc f there s a becto f : V V such that for every { v, v } V, m { v, v } = m f ( v ), f ( v ) () ( ) ({ }). he two multgraphs depcted Fgure 0. are somorphc, uder the becto f ( v ) = u. Fgure 0.: wo somorphc multgraphs.

4 4 Spag ree Results for Graphs ad Multgraphs Edges are cdet at ts ed-odes, ad two odes whch have a edge betwee them are called adacet. Node u has degree deg(u) equal to the umber of edges havg that ode as a ed-ode,.e. =. ({ }) deg( u) m u, v v u he followg theorem, the Frst heorem of Multgraph heory, was proved by Euler 736. heorem 0. (he Frst heorem of Multgraph heory) he sum of the degrees of all the odes of a multgraph M s equal to twce the umber of edges. Proof Let x = vv be a edge of M, the x cotrbutes to both deg( v ) ad deg ( v ). hus whe summg the degrees of the odes of M, each edge s couted twce. Aother way to represet a multgraph s by a matrx. Let M = ( V, m) =,,,. he be a multgraph of order, wth ode set V { v v v } adacecy matrx of M, deoted A ( M ) or smply A s the matrx a where a m( { v, v } ). = the a = deg( v ) =. Note f A s the adacecy matrx of M, Example 0. he adacecy matrx of the multgraph Fgure 0. s

5 A Itroducto to Relevat Graph heory ad Matrx heory A = A multgraph M s r-regular f each ode has degree equal to r. Whe the specfc value of r s ot eeded we say the multgraph s regular. he multgraph M ( V, m ) ( V, m) M = f V V = s called a submultgraph of ad m ({ u, v} ) m( { u, v} ) for all { u, v} V () V = V, the M s called a spag submultgraph. If M s a graph the we use the terms subgraph ad spag subgraph. If ({, }) = ({, }) for all { u, v} V (), the M ( V, m ) m u v m u v. If = s a duced submultgraph ad s deoted by V. If V cotas o edges,.e. m( { u, v }) = 0 for all { u, v} V () the V s called a ( ) { ({ }) } depedet set of odes. If V = V ad m { u, v} = m m u, v, for all { u, v} V (), the M ( V, m ) = s a spag subgraph called the uderlyg graph of the multgraph ( V, m ). I Fgure 0. G s the uderlyg graph of multgraph M. A path a multgraph M s a alteratg sequece of odes ad edges 3 v, x, v, x, v,, x, v, where { v v v },,, are dstct odes ad x s a edge betwee v ad v +. If the multgraph s fact a graph the t s oly ecessary to lst the odes. he legth of a path s the umber of edges the path. It s evdet from the defto that a path o odes has legth. A cycle a multgraph M s a alteratg

6 6 Spag ree Results for Graphs ad Multgraphs sequece of odes ad edges v, x, v, x, v3,, x, v, x, v, where { v v v },,, are dstct odes ad 0 x s a edge betwee v ad v,, ad x s a edge betwee v ad v 0. he legth of a cycle s the umber of edges the cycle, so a cycle o odes has legth. I a multgraph t s possble to have cycles of legth,.e. f there are multple edges betwee a par of odes, but a graph the mmum cycle legth s 3. A graph G s acyclc f t has o subgraphs whch are cycles. A multgraph s coected whe every partto of the ode set V = V V, V, V, ad V V = has at least oe multedge wth oe edpot V ad the other V. Alterately, a multgraph s coected f there s at least oe path betwee every par of odes. A multgraph whch s ot coected s dscoected. A compoet of a multgraph M s a maxmal coected submultgraph M,.e. f M s a submultgraph of M that properly cotas M, the M s dscoected. A dscoected multgraph cotas at least two compoets, thus a multgraph s coected f ad oly f t has oe compoet. A tree s a coected graph wth odes ad edges. Cosderg a logest path a tree t s easy to see that a tree has at least two odes of degree, called leaves or pedat edges. A spag subgraph that s also a tree s called a spag tree. Fgure 0.3 depcts a multgraph ad oe of ts spag trees. It s easy to see that a multgraph has spag trees f ad oly f t s coected.

7 A Itroducto to Relevat Graph heory ad Matrx heory 7 M Fgure 0.3: A multgraph M ad a spag tree. Oe of the purposes of ths boo s to determe whch multgraph wth odes ad e edges,.e. e, spag trees amog all multgraphs e,, has the greatest umber of. Sce spag trees exst oly f the multgraph s coected, gve a coected multgraph M ( Vm, ), the followg procedure wll produce a spag tree. Choose a ode u at radom, set V u, V V V, ad E. If V the sce M s coected there s at least oe edge wth oe edpot V ad the other V. Choose oe such edge, add the edge to E ad remove the edpot from set V ad add t to V. Cotue ths process utl V ad therefore V V, V E s a spag tree.. he resultg graph he process foud above ca be used to fd spag trees but t s effcet for fdg all spag trees of a multgraph; the multgraph Fgure 0.3 has 34 other spag trees. M Vm,, where m s a A drected multgraph s a par oegatve teger-valued fucto defed o V V u, u u V. As for multgraphs, we refer to the elemets of V as odes. A mult-arc s

8 8 Spag ree Results for Graphs ad Multgraphs ( ) m u, v 0 ; a elemet ( u, v) V V { ( u, u) u V} such that (( )) m (( u, v) ) ( u, v ), ( ) s called the multplcty of the mult-arc. If for each par m u v ad m ({ } ) (, ) { 0,}, the ( V, m ) s a drected graph or a dgraph, = A s called the arc-set. We sometmes employ the D = V, A. alterate otato ( ) As the case of multgraphs, drected multgraphs have geometrc represetatos as well. Each elemet of V s depcted by a pot, ad m u, v drected curves, drected from two pots u ad v are oed by (( )) u to v. Fgure 0.4 shows a drected multgraph ad a drected graph. It s tradtoal to refer to each drected curve the represetato as a arc, ad the collecto of all such arcs as A. A arc from u to v s cdet from u ad cdet to v; f such a arc exsts we say u s adacet to v ad v s adacet from u. Node u has degree, deg(u), equal to the umber of arcs cdet to u ad outdegree, outdeg(u), equal to the deg( u) = m v, u ad umber of arcs cdet from u,.e. (( )) =. (( )) outdeg( u) m u, v v u v u M D Fgure 0.4: A drected multgraph M ad a drected graph D.

9 A Itroducto to Relevat Graph heory ad Matrx heory 9 A theorem for drected multgraphs aalogous to the Frst heorem of Multgraph heory follows from the observato that each arc cotrbutes to the degree of a ode ad to the outdegree of aother ode. heorem 0. (he Frst heorem of Drected Multgraph heory) he sum of the degrees of all the odes of a drected multgraph s equal to the sum of the outdegrees, ad these both equal the umber of arcs. Let M = ( V, m) m u v m u v m v u ({, }) = ((, )) + ((, )) { u, v} V() be a drected multgraph ad defe for each two elemet subset. he multgraph M = ( V, m) s called the uderlyg multgraph of M. Alteratvely, f M s a gve multgraph, the ay drected multgraph havg M as ts uderlyg multgraph s referred to as a oretato of M. I Fgure 0.5, M s the uderlyg multgraph of the mult drected graph M, whle M ca be cosdered a oretato of M. M Fgure 0.5

10 0 Spag ree Results for Graphs ad Multgraphs wor. We ow troduce some specal graphs that we wll ecouter ths A complete graph s a graph of the form ( () ) edge betwee every par of odes. If V graph of order by V, V,.e. there s a = we deote the complete K. We depct two complete graphs Fgure 0.6. K 4 K 5 Fgure 0.6: he complete graphs K 4 ad K 5. A subgraph G ( V, E ) = of a graph G = V, E s a clque f t s a maxmal complete subgraph of G,.e. E = V() ad for ay subset of odes V that properly cota V, V s ot complete. I the graph depcted Fgure 0.7, { v, v, v, v } s a clque, whle {,, } complete but ot a clque. 3 4 ( ) v v v s 3 v v v 4 v 3 Fgure 0.7: A clque sde a graph. he clque edges are thcer.

11 A graph A Itroducto to Relevat Graph heory ad Matrx heory G = s bpartte f V ca be parttoed to two sets V ad V such that ay edge E has oe edpot V ad the other V. A bpartte graph s a complete bpartte graph f for every ad every v V, uv E. If V = p ad V = q we deote the complete bpartte graph by K p, q. Whe p =, we refer to the graph K,q as a star. More geerally, a graph G ( V, E) = s -partte f V ca be parttoed to sets V, V,, V, such that ay edge E has oe edpot V ad the other ( V, E) u V V, for some <. Note a -partte graph s bpartte. A -partte graph s a complete -partte graph f for every u V ad every v V, uv E wheever. If a complete -partte graph s regular, the each part has the same order. If V = for, the we deote the complete -partte graph by K.,,, Fgure 0.8 depcts the complete bpartte graph K,3, the star K,4 ad the complete 3-partte graph K,,3. Fgure 0.8: -partte graphs.

12 Spag ree Results for Graphs ad Multgraphs Fgure 0.8: -partte graphs (cotued). Suppose,,..., ad are postve tegers such that < < <. he crculat graph C (,,..., ) s the graph havg ode set V = { 0,,,, } wth ode adacet to each ode ( ± )mod for each,. he,,..., deote the ump szes. If = for all,, we deote the crculat by Fgure 0.9 shows two crculats. C Fgure 0.9: he crculats C 8 ad C (,3). 8

13 A Itroducto to Relevat Graph heory ad Matrx heory 3 Gve a graph G ( V, E) graph ( V, V() E) =, the complemet of G, deoted G, s the,.e. G has the same ode set as G ad uv s a edge of G f ad oly f uv s ot a edge of G. A graph ad ts complemet are show Fgure 0.0. G Fgure 0.0: A graph ad ts complemet. O occaso t s ecessary to delete a set of odes or a set of edges from a multgraph. Let M = ( V, m) be a multgraph wth edge set E. If U V s a set of odes, the M U s the multgraph obtaed by deletg the ode set U ad all edges cdet o a ode U,.e. M U ( V U, m') { u, v} ( V U ) (). If U { u} F = where m' ({ u, v} ) m( { u, v} ) = we wrte M u = for all stead of M { u} E s a set of edges, the, M F s the multgraph obtaed by deletg the edges F, but ot the ed-odes. If x s a edge of M ad F = { x} we wrte M x stead of M { x}.. If Remar 0. here s a ambguous case,.e. M { u, v}, where { u, v } ca be ether the set cosstg of the two odes u ad v or a multedge cdet o the two odes u ad v. If we are deletg the two ode set we

14 4 Spag ree Results for Graphs ad Multgraphs wll wrte M u v ad f we are deletg a edge cdet o u ad v we wll wrte M uv. Remar 0.3 Oe ca also th of the complemet of a graph ( V, E) G = as the graph obtaed by deletg the edge set E from the complete graph o ode set V. Example 0. Cosder the multgraph M, depcted Fgure 0.a, wth specfed odes labeled u ad v ad specfed edges x ad y. he multgraphs M u v 0.c, respectvely. ad M { x, y} are show Fgure 0.b ad x u y v a) A multgraph M b) c) Fgure 0.

15 A Itroducto to Relevat Graph heory ad Matrx heory 5 A set of edges a multgraph s a matchg or a depedet set of edges f o two edges the set are cdet at the same ode. A matchg cossts of edges, whch we deote by K, sce each edge s somorphc to K. We ote that by the defto of a matchg,. A specal case of edge deleto that we wll ecouter s mus a matchg, deoted K K. I Fgure 0. we depct K 5 K. Note, the two edges deleted are arbtrary as log as they are ot cdet o a ode, as all such graphs are somorphc. K Fgure 0.: K5 K. he deleted edges are dcated by the dotted edges. At tmes t may be ecessary to add a edge to a multgraph. Let M = ( V, m) be a multgraph ad u, v V, the uv M + s the multgraph obtaed by addg edges betwee the odes u ad v,.e. M = ( V, m ), where m { u, v } + uv Whe =, we wrte uv M + ({, }) f {, } {, } ({ u, v }) otherwse m u v + u v = u v ( ) = m. If x s a arbtrary edge, the the multgraph wth edges added betwee a par of odes. x M + deotes. Example 0.3 Cosder the multgraph M, depcted Fgure 0.3a, wth specfed odes labeled u, v ad w. he multgraphs uv M + ad

16 6 Spag ree Results for Graphs ad Multgraphs uw M + are show Fgure 0.3b ad 0.3c, respectvely. I Fgure 0.4 x we show K + 5, the placemet of the edge x s arbtrary. u w v a) A multgraph M u u w w b) { u v} v v { }, M + u, w c) M + Fgure 0.3 x x Fgure 0.4: K + 5.

17 A Itroducto to Relevat Graph heory ad Matrx heory 7 If x = uv s a edge of a multgraph M, the the cotracto M x s the multgraph obtaed from coalescg u ad v, the ed-odes of x, ad deletg ay loops whch result. he cotracto results a multgraph wth oe fewer ode ad, m u v fewer edges. Example 0.4 Cosder the multgraph M, depcted Fgure 0.5a, wth specfed edges labeled x ad y. he multgraphs M x ad M y are show Fgure 0.5b ad 0.5c, respectvely. x y a) A multgraph M b) M x c) M y Fgure 0.5: Cotractos. We ed ths secto by troducg some bary operatos o graphs. Let G V, E ad G V, E graphs ad wth V V. he uo of the G, deoted G G, s the graph V V, E E G.

18 8 Spag ree Results for Graphs ad Multgraphs he o of the graphs G ad G, deoted G + G, cossts of G G ad all possble edges betwee the odes G ad those G (see Fgure 0.6b). he product of the graphs G ad G, deoted G G, s the graph ( V V, E ) where {(, ),(, )} u v u v E f ad oly f u = u v v ad E or v = v ad uu E (see Fgure 0.6c). v a) v 3 u v u v v 3 b) K K c) 3 Fgure 0.6: he o ad the product of two graphs.

19 A Itroducto to Relevat Graph heory ad Matrx heory 9 0. Matrx heory Some fudametal but perhaps less well-ow results from matrx theory to be used the sequel are cluded wth proofs for the sae of completeess. We assume that the reader s famlar wth basc matrx deftos ad operatos, such as sze of a matrx, matrx addto, matrx multplcato, scalar multplcato traspose, ad determat. For deftos of these ad ay other cocepts ot explctly defed here we refer the reader to Lacaster ad smeetsy [Lacaster, 985]. We wll frst expla some matrx otato that wll be used. If P s a matrx of sze r s, the p p p s p p p pr pr prs s P = = p. If the sze s uderstood we wll suppress the subscrpt the latter. We refer to a matrx of sze s or r (.e., a row matrx or a colum matrx, respectvely) as a vector. Uless explctly stated, all vectors are colum matrces ad we wrte matrx, r r s x x x =. Fally, I r deotes the r r detty x O r s deotes the r s matrx zero matrx wth all etres 0, 0 r the zero vector wth all etres 0, ad r the vector wth all etres. If the sze s uderstood we suppress the subscrpts. We ote that all matrces ad vectors are bold-faced.

20 0 Spag ree Results for Graphs ad Multgraphs he frst result we preset cocers the determat of a product of two matrces. heorem 0.3 he Bèt-Cauchy heorem [Lacaster, 985]. If P ad Q = q are matrces over a arbtrary feld, wth = p l l, the det(pq) = l det P det < <... < l Q, 3 p where the mor C 3 deotes the submatrx of C p obtaed by deletg all rows except,,, p ad all colums except,,, p. I other words, the determat of the product PQ s equal to the sum of the products of all possble mors of (the maxmal) order of P wth correspodg mors of Q of the same order. Proof Observe that

21 A Itroducto to Relevat Graph heory ad Matrx heory l l l pτ q τ pτ q τ pτ q τ τ = τ = τ = l l l pτ q τ pτ q τ p q τ τ PQ =. τ = τ = τ = l l l pτ q τ pτ q τ pτ q τ τ = τ = τ = Now, by successvely applyg the learty of the determat for a specfc colum, we obta p q p q p q l l l p q p q p q det( PQ) det τ = τ = τ = p q p q p q τ τ τ τ τ τ τ τ τ τ τ τ = τ τ τ τ τ τ whch s a sum of l determats. Of course, f a b τ = τ, where a b, the the correspodg term the sum vashes ad the sum reduces to oly those terms for whch τ, τ,, τ are dstct. Now cosder a specfc lst < < < l, ad usg σ to deote permutato, t follows that det ( PQ ) = < <... < l σ (,,..., ) = ( τ, τ,..., τ ) p q p q p q p q p q p q p q p q p q τ τ τ τ τ τ τ τ τ τ τ τ det. τ τ τ τ τ τ But

22 Spag ree Results for Graphs ad Multgraphs p q p q p q p q p q p q det p q p q p q τ τ τ τ τ τ τ τ τ τ τ τ = τ τ τ τ τ τ p q p q p q I p q p q p q σ det, p q p q p q ( ) where I σ deotes the umber of versos the permutato (,,, ) = (,,, ) σ τ τ τ learty of determats for a fxed colum,. Aga, successvely applyg the Iσ q q q < <... < l σ (,,..., ) = ( τ, τ,..., τ ) det( PQ) = ( ) det P. Fally, the result follows from the observato that Iσ det Q = ( ) q q q. σ Example 0.5 Let Bèt-Cauchy: 4 0 P = 3 ad Q =, the applyg 3

23 A Itroducto to Relevat Graph heory ad Matrx heory 3 det( PQ ) 4 0 det det det det det det Multplyg the matrces, PQ = ad det Before we proceed wth the remader of ths secto, we wll revew some pertet materal from Lear Algebra. A set of vectors a subspace of f the followg three codtos hold: () W s 0 W, () x y W for every xy, W, ad () cx W for every c, x. Gve a set of vectors x,x,,x the spa of the vectors, spa x,x,,x s the set cosstg of all lear combatos of the vectors,.e. c c c c c c spa x,x,,x x x x,,,. It s a easy exercse to show that spa x,x,,x s a subspace of. A set of vectors x,x,,x s learly depedet f cx c x c x 0 oly whe c 0, for all < <. If the set of vectors s ot learly depedet, the t s learly depedet,.e. there exst c, c,, c at least oe of whch s ot 0 such that cx c x c x 0. A bass of a subspace W s a learly depedet set of vectors that spa W. he dmeso of a subspace W, dm W, s the umber of vectors ay bass of W. Gve two vectors

24 4 Spag ree Results for Graphs ad Multgraphs x x x = ad x the otato y y y =, the dot product y x y = = x y. We wll also use x y. wo vectors x ad y are orthogoal f 0 he orthogoal complemet of a subspace { y R y x = 0, for all x W} a vector, e x y =. W R, W =. he Eucldea orm or ust orm of x or smply x, s gve by x = x x. A set of vectors e { x,x,,x } s orthoormal f the vectors each have orm ad are parwse orthogoal,.e., x x = 0, =. Ay set of orthoormal vectors s ecessarly learly depedet. A orthoormal bass s a bass whose vectors form a orthoormal set. Let A be a matrx. A egevalue of A s a scalar λ such that Ax = λx for some o-zero vector x. Ay o-zero vector x such that Ax = λx s a egevector assocated wth the egevalue λ. he characterstc polyomal of A, P ( λ) = det ( λ ) the roots of the equato PA ( λ ) = 0. A I A. Egevalues are Proposto 0.4 If A s a real symmetrc matrx the t has real egevalues ad a orthoormal bass of assocated egevectors. Proof Suppose A s a real symmetrc matrx, W s a subspace of R ad W s varat uder A ad A ( W ) W. If { x x x } A,.e. ( W ) A W ad,,, s a orthoormal bass for W the there

25 A Itroducto to Relevat Graph heory ad Matrx heory 5 exsts uque real matrces B ad C such that f = = Ax β x, ad = = A x χ x, the χ α = C. It s easy to see that χ α ad A ( W ) W, the A ( ) = = x α x, β α = B ad β α B = C so that f A s symmetrc W W ad B s symmetrc. Our frst clam s that f A s symmetrc the the lear operator x Ax has real egevalues W (cludg multplctes). Ideed, the egevalues of B are those of x Ax. Now det ( λi B ) splts to lear factors over the complex umbers ad therefore has roots (cludg multplctes) whch are readly see to be real. Next we clam that each subspace W of dmeso whch s varat uder A has a orthoormal bass of egevectors { y y y },,, correspodg to the egevalues λ, λ,, λ. We prove ths by ducto o. If = the W s spaed by the ut vector y ad sce W s varat uder A we have that Ay = λy,.e. λ s a real egevalue ad { y } s the orthoormal bass. Now suppose ad the clam s true for all varat W of dmeso <. Let dm(w) = ad ( W ) A W. Let y W be a egevalue of orm oe assocated wth the real egevalue λ. he spa{ } W = y s varat uder A. But the W W, the collecto of all vectors W orthogoal to W, s also varat uder the symmetrc matrx A,.e. f x W the

26 6 Spag ree Results for Graphs ad Multgraphs ( y A) x y ( Ax ). Sce ( ) 0 = = dm W W =, there exsts real egevalues ad assocated orthoormal egevectors y, y3,, y for A o W W, thereby establshg the clam. Fally we may apply ths clam to = sce A( R ) R. heorem 0.5 (Courat-Fscher Max-M heorem) [Lacaster, 985] If the egevalues of the symmetrc matrx A are amed order of creasg value,.e. λ λ λ, the λ = max m L vares over all ( ) L + -dmesoal subspaces of x Ax 0 x L x x R. where o prove the Courat-Fscher heorem, we frst requre a Lemma. Lemma 0.6 For each let dmesoal subspace of egevalues λ λ λ. he L deote a arbtrary ( ) + - R, ad cosder a symmetrc matrx A wth λ x Ax m 0 x L x x x Ax ad λ + max 0 x L x x. Proof (of Lemma 0.6) Let x,x,,x be a orthoormal system of egevectors for the correspodg egevalues λ λ λ of A,.e., Ax = λx, ad let L ˆ = spa{ } x,x,,x for =,,,. Fx, ad observe that, sce dm( ) dm( ˆ ) ( ) L + L = + + >, there

27 A Itroducto to Relevat Graph heory ad Matrx heory 7 exsts a ozero vector x0 partcular, sce x 0 L, t follows that x0 ˆ R belogg to both of these subspaces. I = = α x ad thus, Hece, x0 Ax0 = αx A αx = λ α = = =. x0 Ax 0 = = λ α λ α x Ax, or λ x x λ. he frst result must follow because x 0 L, ad so m 0 x L x Ax x Ax λ. x x x x 0 0 he secod result s a cosequece of replacg L {,,..., } spa x x x. by Proof (of Courat-Fscher) Because of Lemma 0.5, t suffces to show the exstece of a ( +)-dmesoal subspace such that m ( Ax, x) 0 x L such a subspace. = λ Ideed, tag L. spa{ x, x +,..., x} =, we obta A matrx A s postve sem-defte f x Ax 0 for all vectors x. Note: a matrx wth all ts prcpal mors o-egatve s postve semdefte. A matrx A s postve defte f x Ax > 0 for all vectors x 0.

28 8 Spag ree Results for Graphs ad Multgraphs Proposto 0.7 If A s a postve sem-defte real symmetrc matrx, the ts egevalues satsfy λ λ λ. If A s postve defte, 0 the ts egevalues satsfy < λ λ λ. 0 Proof Let A be a postve sem-defte real symmetrc matrx. We ow from Proposto 0.4 that all λ are real. Next cosder where Ax = λx, x 0, so that 0 x Ax ad hece λ 0. he result for postve defte matrces follows by replacg the equalty 0 x Ax wth 0 < x Ax. We refer to the lst of egevalues of a matrx as the spectrum of the matrx. Corollary 0.8 Suppose that A s a postve sem-defte real symmetrc matrx, ad B s a real symmetrc matrx. Let α, α,, α deote the egevalues of A + B ad β, β,, β deote the egevalues of B. he α β for all =,,,. Proof Note that ( ) + = + x A B x x Ax x Bx x Bx so that for each =,,,. x ( A + B) x x Bx α = max max = β, ad L x x L x x ( + ) x A B x α = max m L 0 x L x x x Bx max m = m max x Bx = β L 0 x L x x y = x = x y =, 0 e e = +,... = +,...

29 A Itroducto to Relevat Graph heory ad Matrx heory 9 he followg Lemma, the cotrapostve of the Levy-Desplaques- Haddamard heorem [Lacaster, 985], presets a useful property of sgular matrces: heorem 0.9 (Domat Matrx heorem) If A s a sgular matrx,. the there exsts a row such that a a Proof If a matrx A s sgular, the there s a o-zero vector x so that Ax = 0. Hece, a x = 0 for all, or a x = a x for all, ad a x a x,. Clearly,, ( max ) a x x a for all. Let ow be chose so that x s the largest value of x for =,,, ; hece, max x x = max x for all. Further, sce x 0, x = max x > 0. herefore, a a. Cosder the matrx A = M λi whch s sgular for ay λ, a egevalue of M. Applyg heorem 0.7 we see that there exsts a such that m λ = λ m m. Now, usg the secod tragle equalty x y x y, we get m λ λ m m ad λ m λ m m. Sce these equaltes are true for each egevalue λ, ad some, t follows that

30 30 Spag ree Results for Graphs ad Multgraphs m max m m λ λ m. m max + m Of course, f we are dealg wth postve sem-defte matrces, the λ = λ. hs leads to the followg corollary: Corollary 0.0 (Gersgor s heorem) Gve a -square complex matrx M m we have egevalue bouds = λ m m m m ad λmax max m m. Our ext result gves a relatoshp betwee egevalues of a square matrx to the trace ad to the determat of the matrx. heorem 0. Let A be a matrx, wth egevalues λ, λ,, λ, the λ = trace( A ) ad λ = det ( ) = Proof Let = A. = A a be a matrx ad P ( λ) = det ( λ ) the characterstc polyomal of A. Sce P ( λ) A I A be A s a polyomal of degree we ca wrte t the form the ( λ) = λ λ + + ( ) P p p A 0. From ths form we see that (0) ( ) A = p0. But pluggg 0 to P ( λ) = det ( λ ) P A ( 0) det ( A) ( ) det( A ), thus p 0 = det ( ) P = = ( I ) A I A we get A. Now evaluatg det λ A ad collectg le terms we see that the coeffcet of λ, p a a a trace. If λ, λ,, λ are, s ( ) = ( A)

31 A Itroducto to Relevat Graph heory ad Matrx heory 3 the egevalues of A,.e. the roots of PA ( λ ) = 0, the we ca also express PA ( λ ) the form P ( λ) = ( λ λ )( λ λ ) ( λ λ ) Multplyg out we get ( ) = ( ) + + ( ) A. PA λ λ λ λ λ λ λ λ λ. Sce the coeffcets of the polyomal are uque, the results follow from equatg the coeffcet of expasos. λ ad the costat term both he followg developmet of some algebrac graph theoretcal deas wll be useful the sequel. If B ad C are ad m l matrces over a arbtrary feld, the the Kroeecer product of B ad C, B C, s the m l matrx wth bloc form bc b C b C b b b C C C. bc b C bc he Kroeecer sum of a matrx B ad m m matrx C, deoted B C, s smply B Im + I C. Example = =

32 3 Spag ree Results for Graphs ad Multgraphs ad = 3 I + I = I I = I I =

33 A Itroducto to Relevat Graph heory ad Matrx heory heorem 0. (A Assocatve Law of Products) If the products EF ad UV are defed, the ( EF) ( UV) = ( E U)( F V). Proof Wthout loss of geeralty, suppose that EF s m ad UV s ( ) ( ) αβ ( ) pq l. he ( EF) ( UV) = EF UV where = ( α ) l + p wth p l, α ; ad q, β m. O the other had, f E has s colums ad U has t colums, the ( αγ )( γβ ) (( E U)( F V) = ( E U) ( F V) = E U F V, z z pr rq z z where ad are as above, ad ( γ ) s z = m + r wth γ s, r t. Hece, = ( αγ γβ ) ( ) = ( ) ( ) αβ ( E U )( F V ) E F U V EF UV. t pr rq pq γ = r= A mmedate cosequece cocers the verse of a Kroeecer product: Corollary 0.3 (Iverse of a Kroeecer Product) If P ad Q are vertble, the so s P Q = P Q. P Q ad ( )

34 34 Spag ree Results for Graphs ad Multgraphs Proof Cosder ( P Q)( P Q ) PP QQ Im I Im = = =, where P s m m ad Q s. heorem 0.4 (Egevalues of a Kroeecer Polyomal) Let B ad φ( B; C) = α B C a fte sum. C be square matrces, ad set ( ), he the egevalues of φ( B; C) are gve by = φ( β ; γ ) α β γ, where β ad γ ru through the egevalues of B ad C, respectvely., Proof Sce the etres of B ad C are complex, there exst vertble matrces P ad Q such that B = PBP ad C = QCQ are upper tragular. Of course, = B PB P ad = tragular, ad therefore so s each matrx ( α ) sum α ( B C ), 0.3 that C QC Q are also upper B C as well as the. Now t follows from heorem 0. ad Corollary ( B C ) = P B P Q C Q = ( P Q) ( B C ) = ( P Q) φ( B ; C ) = α B C = P Q whch s equal to ad thus ( ) ( ), ( ) ( ) = ( ) φ ( ; )( ) α B C P Q P Q B C P Q.,

35 A Itroducto to Relevat Graph heory ad Matrx heory 35 herefore, φ ( B ; C ) ad ( ; ) ( ; ) φ B C have the same egevalues, but φ B C s upper tragular wth dagoal etres gve by = αβ γ as β ad, ( ; ) φ β γ γ vary through the egevalues of B ad C, respectvely. Hece, the result follows. Corollary 0.5 If B ad C are square matrces, the the egevalues of B C ad B C are gve by βγ ad β + γ, respectvely, as β ad γ vary through the egevalues of B ad C. Proof that hs result follows from heorem 0.4 after observg B C ad 0 0 ( ; ) B C are obtaed from ( ; ) φ b c = bc ad p φ s b c = b c + bc, respectvely, upo substtuto of B for b ad C for c, ad the replacemet of ordary multplcato by Kroeecer product.