Trace of Positive Integer Power of Adjacency Matrix

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1 Global Joual of Pue ad Appled Mathematcs. IN Volume, Numbe 07), pp Reseach Ida Publcatos Tace of Postve Itege Powe of Adacecy Matx Jagdsh Kuma Pahade * ad Mao Jha Depatmet of Mathematcs, Maulaa Azad Natoal Isttute of Techology, Bhopal M.P.) Ida. Depatmet of Mathematcs, Maulaa Azad Natoal Isttute of Techology, Bhopal M.P.), Ida. Abstact Fdg the tace of postve tege powe of a matx s a mpotat poblem matx theoy. I the Gaph Theoy a mpotat applcato of the tace of postve tege powe of Adacecy matx s coutg the tagles coected gaph. I ths pape we peset a ew effcet fomula to fd the tace of postve tege powe of some specal Adacecy matx of coected smple gaph. The key dea of ou fomula s to multply the matx k tmes, whee k s postve tege. Keywods: Adacecy matx, Complete Gaph, Tace, Matx multplcato, Matx powe. INTRODUCTION Taces of powes of matces ase seveal felds of mathematcs. Moe specfcally Netwok Aalyss, Numbe theoy, Dyamcal systems, Matx theoy, ad Dffeetal equatos []. Thee ae may applcatos matx theoy ad umecal lea algeba. Fo example, ode to obta appoxmatos of the smallest ad the lagest Ege values of a symmetc matx A, a pocedue based o estmates of the tace of A ad A -, s atual, was poposed []. Whe aalyzg a complex etwok, a mpotat poblem s to compute the total umbe of tagles

2 080 Jagdsh Kuma Pahade ad Mao Jha of a coected smple gaph. Ths umbe s equal to TA )/ whee A s the adacecy matx of the gaph [].It s possble to solve tagle fdg ad coutg a gaph by adacecy matx epesetato of the gaph [9]. Fo ay tege matx M ad ay pme umbe p, the etes of the uque Wtt vecto cosstg of p-adc teges ae expessed fom the taces of powes of the tege matx M [].Taces of powes of tege matces ae coected wth the Eule coguece [] p TM p T M mod p ) Holds fo all tege matces M, all pmes p, ad all atual. Thee ae may applcatos of ths coguece to dyamcal systems, whch s atual studyg those vaats of dyamcal systems that ae descbed tems of the taces of powes of tege matces, fo example studyg the Lefschetz umbes [].oluto of Lyapuov matx equatos ca be solved by usg matx polyomals ad chaactestc polyomals whee the computato of the taces of matx powes ae eeded [].Thee s a questo [],ca tace k Awhee k A s the k-th symmetc powe of matx A) be wtte tems of some coeffcet of A k+ ad the coespodg coeffcet A? The computato of the tace of matx powes has eceved much atteto. I [], a algothm fo computg TA k ), k s tege, s poposed, whe A s a lowe Hessebeg matx wth a ut codagoal. I [7], a symbolc calculato of the tace of powes of Tdagoal matces s peseted. Tace of postve tege powe of eal matces s peseted [8].A fomula fo tace of ay symmetc powe of a matx s peseted [0] tems of the oday powe of matx. Let A be a symmetc postve defte matx, ad let {λk} deote ts Ege values. Fo q R, A q s also symmetc postve defte, ad t holds [] TA q =Ʃkλk q But fo hghe ode matx fdg Ege values λk s vey dffcult ad tme cosumg, theefoe othe fomula to compute tace of matx powe s eeded. Now we peset a ew theoem to compute tace of matx powe fo adacecy matx of a coected smple gaph wth ay umbe of vetces. Ou estmato fo the tace of A k s based o the multplcato of matx. Ths fomula wll deped oly o ode of the matx. Tace of a matx A = [a], s defed to be the sum of the elemets o the ma dagoal of A..e. TA = a + a +..+a A complete gaph s a smple udected gaph whch evey pa of dstct vetces s coected by a uque edge. Fo a gve umbe of vetces, thee's a

3 Tace of Postve Itege Powe of Adacecy Matx 08 uque complete gaph. The adacecy matx of the complete gaph takes the patculaly smple fom of all ˈs wth 0ˈs o the dagoal,.e., the ut matx mus the detty matx. MAIN REULT Theoem.Let A s a adacecy symmetc matx of a complete smple gaph wth vetces, the TA ad k / / s k, ) k k, fo eve postve tege k k TA s k, ), fo odd postve tege k. whee k, be a umbe depedg upo k ad, ad defed by k k,, k, k /, k, k /, ad, k, k, k, Poof: Cosde a adacecy symmetc matx A = a),, whee a = 0 f f Now A = a),, whee a = f f HeceTA = O TA. =, O TA = s, ) ) ).

4 08 Jagdsh Kuma Pahade ad Mao Jha Aga A = a),, whee a = f f Hece TA = O TA =., O TA =. ) ) ), Aga A = a),, whee a = f f O a = f f Hece TA = O TA =.. ) ),) ) ), s O TA =. ) ) ), Aga A = a),, whee a = f f

5 Tace of Postve Itege Powe of Adacecy Matx 08 O a = f f Hece TA = O TA =.. ) ),) ) ), O. ) ) ), T A Aga A = a),, whee a= f f O a = f f Hece TA = ) + ) ) + ) O TA =.. ) ),) ) ), +. ) ),) O TA =. ) ) ), Aga A 7 = a),, whee a= f f

6 08 Jagdsh Kuma Pahade ad Mao Jha O a f f Hece TA 7 = ) ) + ) ) ) + O TA 7 = , ) ) 7,) ) + 7,) ) ) 7. ) O TA 7 = 7, ) ) ) 7. Cotug ths pocess of multplcato of matx, we coclude that, TA k / s k, ) k, fo eve postve tege k ad / k TA whee k k k,, be a umbe depedg upo k ad, ad defed by k, fo odd postve tege k. k,, k, k /, k, k /, ad, k, k, k, Ths completes the poof.

7 Tace of Postve Itege Powe of Adacecy Matx 08 Example : Cosde a matx A = TA. 0 0 [ ] ad let we ae to fd Hee = ad k =.the by ou theoem, we have TA =, ) ) ). O TA =.., ) ),) ) O TA = + = 0 Example : How may tagles ae thee the followg gaph? ) We kow that the umbe of tagles the coected gaph s TA )/, whee A s the adacecy matx of the gaph. The adacecy matx of ths gaph s A = By ou theoem TA =, ) ) ) Hee =, TA =,) ) )..

8 08 Jagdsh Kuma Pahade ad Mao Jha O TA = = Numbe of tagles the gaph = TA )/ = / =,amely,,,. Example : How may tagles the followg gaph? 7 Hee adacecy matx of the gaph s A = a), 7, 7 whee a 0 f f By ou theoem TA =, ) ) ) Hee = 7, TA. =,)77 ) 7 ) O TA = 7 = 0 Numbe of tagles the gaph = TA )/ = 0/ =.. CONCLUION AND FUTURE WORK I ths pape,we exteded the ode of matx peseted [8] fo the tace of postve tege powe of eal matx to the tace of postve tege powe of some adacecy matx of ay ode.as explaed the pape such estmates have applcatos vaous baches of mathematcs. The deas peseted ths pape could be exteded to ay matx.

9 Tace of Postve Itege Powe of Adacecy Matx 087 ACKNOWLEDGMENT We would lke to hadly thakful wth geat atttude to Decto, Maulaa Azad Natoal Isttute of Techology, Bhopal fo facal suppot ad we also thakful to HOD, Depatmet of Mathematcs of ths sttute fo gvg me oppotuty to expose my eseach scetfc wold. REFERENCE [] B. N. Dattaad K. Datta, A algothm fo computg powes of a Hessebeg matx ad ts applcatos, Lea Algeba Appl., 97), pp [] H. Avo, Coutg tagles lage gaphs usg adomzed matx tace estmato, Poceedgs of kdd-ldmta 0, acm, 00. [] Claude Bezsk, Paaskev Fka, ad Malea Mtoul, Estmatos of the tace of powes of postve self-ado opeatos by extapolato of the momets, Electoc tasactos o umecal aalyss, Volume 9, pp. -, 0. [] V. Pa, Estmatg the extemal egevalues of a symmetc matx, Comput. Math. Appl.,0990), pp. 7. [] N. Hgham, Fuctos of Matces: Theoy ad Computato, IAM, Phladelpha, 008. [] A. V. Zaelua, O cogueces fo the taces of powes of some matces, Poc. teklovist. Math., 008), pp [7] M. T. Chu, ymbolc calculato of the tace of the powe of a tdagoal matx, Computg, 98),pp [8] Pahade, J. ad Jha, M. 0) Tace of Postve Itege Powe of Real Matces. Advaces Lea algeba & Matx Theoy,,0-. [9] Matheu Latapy, Ma-memoy tagle computatos fo vey lage gaph, Theoetcal Compute cece ) 8 7. [0] Cseos J.L.,Heea R. ad ataa N.,A fomula fo the tace of symmetc powe of matces, math DG 0,aXv:.0v. [] Cseos-Mola, J.L.,A vaat of matces, Electo.J.Lea Algeba 00),-.

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