International Journal of Scientific & Engineering Research, Volume 6, Issue 7, July ISSN

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1 Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July ISSN HARATERISTIS OF EDGE UTSET MATRIX OF PETERSON GRAPH WITH ALGEBRAI GRAPH THEORY Dr. G. Nirmala M. Murugan 2. Principal, Govt Arts ollag Mlur, Mudurai (D.T) 2. Rsarch Scholar Dpartmnt of Mathmatics, Priyar Maniyammai Univrsity Vallam Thanjavur. Abstract: In this work Basic concpts of algbraic graph thory and its proprtis ar rviwd and xtndd to th rlatd concpts of dg cutst matrix in Ptrson graph and its proprtis. Th rlation btwn dg cutst matrix and incidnc matrix is Introducd Rank of th Ptrson graph dg cutst matrix is also rviwd. Ky Words: Ptrson graph incidnc matrix dg cutst matrix Rank of th Ptrson graph. Introduction Algbraic graph thory can b viwd as an xtnsion to graph thory. In Which algbraic mthods ar applid to graph thory problm. Edg cutsts ar of grat importanc in studying proprtis off communication and transportation ntworks. Th ntwork nds strngthning by mans of additional tlphon lins. All cut sts of th graph and th on with th smallst numbr of dgs is th most valuabl. This papr dals with Ptrson graph and its Dfinition 4 proprtis with cut-st matrix and diffrnt cut sts in a A connctd graph G is a dg cutst is a st of Ptrson graph. Rlation btwn dg cutst matrix dgs whr rmovd from G lavs G is disconnctd. with incidnc matrix ar xplaind. Rank of th dg Providd rmoval of no propr subst of ths dgs cutst matrix in a Ptrson graph is dalt with disconnctd G. A dg cutst always cuts a graph into two Dfinition Th dgr of a point vi in a graph G is th numbr of lins incidnt with vi is dnotd by dg (v i) A point v of dgr o is said to b isolatd point. A point v of dgr is said to b pndnt vrtx. Dfinition 2 For any graph G w dfin δ(g) = Min { dg v / v V (G) } (G) = Max { dg v / v V (G) } If all th points of G hav th sam dgr thn δ (G) = (G)= r Thn G is said to b a rgular graph of dgr r Ptrson graph is a rgular graph of dgr Edg cutsts of G ar = { 4} Dfinition Ptrson graph is a - rgular graph of vrtics and 5 dgs. 25

2 Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July ISSN Lt th graph G hav m dgs and q b th numbr diffrnt cut sts in G. 2 = {,2} = {2,} Th dg cutst matrix (G) is givn by (G) = (ij)q x m th th ij = If i cutst includ j dg othrwis ut st matrix of th Ptrson graph ut st matrix of Ptrson graph. Th diffrnt cut sts of th Ptrson graph ar namly,2,.54, 55 4 = {,} 5 = {5,6} 6 = {6,7} 7 = {5,7} Dfinition 5 = {, 6, 8} 2 = {, 2, 7} = {2,, 2} 4 = {,, 4} 5 = {4, 5, } 6 = {5, 6, 9} 7 = { 2, 5, 9} 8 = {, 4, 5} 9 = {8, 4, } = {7,, } = {6, 8, 7,2} 2 = {, 7, 2,} = {2, 2,,4} 4 = {,,,5} 5 = {4,, 9,6} 6 = {, 8, 9,6} 7 = {6, 5, 5,2} 8 = {, 6, 4,} 9 = {, 2,,} 2 = {2,, 5,9} 2 = {, 4, 4,8} 22 = {4, 5,,7} 2 = {2, 9,, 4} 24 = {5,, 8, } 25

3 Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July ISSN = {7,, 4, 5} 26 = {6, 8, 7, 2, } 27 = {, 7, 2,, 4 } 28 = {2, 2,,, 5 } 29 = {,,, 9, 6 } = {4,, 9, 8, } = {5, 9, 8, 7, 2} 2 = {6, 8,,, 2} = {, 7,, 5, 9 } 4 = {2, 2, 4, 4, 8 } 5 = {, 4, 5,, 7 } 6 = {4,, 6, 5, 2 } 7 = {5, 9, 4, } 8 = {, 2, 4,, 5 } 9 = {2,, 9, 4, } 4 = {, 4, 8, 5, } 4 = {4, 5, 7, 5, 4 } 42 = {5, 6, 4,, 2 } 4= {6,,,, 5 } 44 = {, 7, 4, 9, 2 } 45 = {, 8,, 9, 2 } 46 = {5, 8,,, 7 } 47 = {5, 6, 5, 2, } 48 = {, 2,, 4, 5 } 49 = {, 6, 4, 4, } 5 = {6, 8, 7, 2,, 4 } 5 = {, 7, 2,,, 5 } 52 = {2, 2,,, 9, 6 } 5 = {,,, 9, 8, } 54 = {4,, 9, 8, 7, 2 } 55 = {5, 9, 8, 7, 2, } 56 = {6, 8, 2,, 4, 5 } 57 = {6, 8, 2,, 4, 5 } 58 = {6, 8, 7,, 9, 5 } 59 = {, 7,, 5, 6, 5 } 6 = {, 7,, 9 4, } 6 = {, 7, 2, 4 4, 8 } 62 = {2, 2, 4, 8 5, } 6 = {2, 2, 4, 4, 6, } 64 = {2, 2,, 5, 7 } 65 = {,, 5,,, 2 } 66 = {, 4, 5, 7 4, 5} 67 = {,,, 6, 5, 2 } 68 = {4,, 6, 5, 2, } 69 = {4,, 6, 4, 2 } 7 = {4,, 9,, 4, } 7 = {5, 9,, 4, 4, } 72 = {5, 9,,,, 5 } 7 = {5, 9, 8, 2,, } 74 = {5, 8, 4,, 7, } 75 = {,, 9, 5,, } 76 = {, 5, 2, 5, 4, } 77 = { 9, 2, 8,, 7, } 78= { 6, 8, 7, 2,,, 5 } 79= {, 7, 2,,, 9, 6 } 8= { 2, 2,,, 9, 8, } 8= {,,, 9, 8, 7, 2 } 82= { 4,, 9, 8, 7, 2, } 8= { 5, 9, 8, 7, 2,, 4 } 84= { 6, 8, 7, 2, 4, 4, } 85= { 6, 8, 7,, 5, 5 } 86= { 6, 8, 2,, 4, 9, } 87= { 6, 8, 7,,, 4, 9 } 88= { 6, 8, 2, 5,,, } 89= { 6, 8, 2,, 4, 9, 2 } 9= { 6, 8, 2,, 5,, } 9= {, 7, 2,, 5, } 92= {, 7, 2, 4,, 5, 8 } 9= {, 7, 2, 4, 4, 6} 94= {, 7,, 5, 5, 8 } 95= {, 7,, 5, 6,, 4} 96= {, 7,, 8,, 9, } 97= {, 7,, 9,4, } 98= { 2, 2,,, 6, 5, } 99= { 2, 2,, 5, 7, 4, 5} = { 2, 2,, 5,, } = { 2, 2, 4, 8, 5, 7, } 2= { 2, 2, 4, 8,, 9, } = { 2, 2, 4, 4,, 9, 5} 4= { 2, 2, 4, 4, 6, 7, } 5= {,,, 9,, 4,} 6= {,,, 6, 4,, 2} 25 7= {,,, 6, 5, 2} 8= {,, 5,, 2, 6, 8} 9= {,, 5,,, 2} = { 4,, 9, 8, 2,, } = { 4,, 9,, 4, } 2= { 4,, 9,,, 4, } = { 4,, 6, 5,,, 7} 4= { 4,, 6, 5, 2,, } 5= { 4,, 6, 2,, 8, } 6= { 4,, 6, 2, 4, 7} 7= { 5, 9, 8, 7,, 5, } 8= { 5, 9, 8, 2,, 4} 9= { 5, 9, 8, 2,, 4, 5} 2= { 5, 9,, 4,,, } 2= { 5, 9,, 4, 4, 2, 2} 22= { 5, 9,,,, 2 } 2= { 5, 9,,, 5,, 7} 24= {,, 5, 7, 5, 8, } 25= {,, 5, 7, 4, 9, 2} 26= { 5, 6, 2, 8,, 7, } 27= { 2,, 9, 8, 4, 7, } 28= { 6,,, 9, 2,, 7}

4 Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July ISSN = {, 2,, 9, 2,, 8} = { 5, 4, 7, 8, 4, 9, 2} = {, 4, 8, 7,, 9, 2} 2= { 6,,,, 9, 2, } = { 6,,,, 2, 5, } 4= { 6,,, 5,, 2} 5= { 6,,, 5, 7, 4, 5} 6= { 6,, 4,, 5,, 7} 7= { 6,, 4, 4, 2, 5, 9} 8= {, 2, 5,, 8, 4, } 9= {, 2,, 4, 9, } 4= {, 2,, 4, 2, 6, 5} 4= {, 2, 5, 5,, 4, 8} 42= {, 2,, 4, 6, 2, 5} 4= { 2,, 9, 4, 7, 5, 4} 44= { 2,, 9,, 8, 4 } 45= { 2,, 5, 5,, 4, } 46= { 2,, 5, 6, 4,, 7} 47= {, 4, 8,, 2, 6, 5} 48= {, 4, 8, 5, 7, 5 } 49= {, 4, 4,, 5, 5, 2} Exampl = {, 4, 4, 6, 2,, } 5= { 4, 5, 7, 4, 2, 6} 52= { 4, 5,,,, 5, 9} 5= { 4, 5,, 2, 6, 4, } 54= { 6, 5, 5,,,, } 55= { 6, 5, 5, 2, 4, 4, 8} Th dg cutst matrix of th Ptrson graph. Whos ordr is 55 x 5 is dnotd by (G) ( G) = Rsults ) Th prmutation of rows (or) columns in a dg cutst matrix orrsponds simply to rnaming of th dg cutst and dgs rspctivly. 2) Each row in (G) is a dg cutst vctor ) A column with all zros corrsponds to an dg forming a slf loop. Exampl 4 2 G Th graph G has diffrnt cut sts Namly = {2, } 2 = {, 4} = {2, 4 } Th dg cutst matrix of G is givn by (G) = In a dg cutst matrix all th lmnts in th first column has zros and th corrsponding dgs ar slf loops. From this it is clar that Ptrson graph has no slf loop. 4) Paralll dgs from idntical columns in a dg cutst matrix Graph G2 Th graph G2 has diffrnt cut sts namly = {, 2, } 2 = {, 2, 4} = {, 4} Edg cutst matrix of G2 is givn by (G2) = In a dg cutst matrix first two columns ar th sam. orrsponding dgs, 2 ar paralll dgs. v) In a non sparabl graph sinc vry st of dgs incidnt on a vrtx is a dg cutst. Thrfor vry row of incidnc matrix I (G) is includd as a row in th dg cutst matrix (G) That is for a non sparabl graph (G) ontains I (G) 4 25

5 Intrnational Journal of Scintific & Enginring Rsarch, Volum 6, Issu 7, July ISSN For a sparabl graph th incidnc matrix of a ach Block is containd in th dg cutst matrix. matrix of th Ptrson graph has similarity with th rgular graph proprtis of th dg cutst matrix. Thorm Rfrncs: If G is a connctd graph thn th rank of a dg cutst matrix (G) is qual to th rank of th incidnc matrix I (G) which is qual to th rank of graph G.. 2. H.J. Finck, on th chromatic Numbr of Graph and its complmnts Thory of Graphs procdings of th colloquium, Thihany Hungaru 996,99 () R. Balakrishnan and K. Rnganathan A txt Book Proof Lt I (G), B (G), (G) b th incidnc, ycl, ut. of Graph thory, Springr 2. G.Nirmala and D.R Kirubaharan Uss of lin st matrix of th connctd graph G, Thn w hav graphs Intrnational journal of Humanitis Rank of (G) n-. ( scincs PMU_Vol 2-2. ) Sinc th numbr of dgs common to a dg cutst and a cycl is always vn. Evry row in is orthogonal to vry row in B. Providd th dgs in both. Band ar arrangd in th G. Nirmala and S. Priyadharshini Fuzzy algbra and Groups, Intrnational Journal of omputrs, Mathmatical Scincs and applications Vol - 2 G. Nirmala and S. Priyadharshini Q- ut with P- sam ordr. Fuzzy algbra procdings of Intrnational Thus B T = B T (Mod 2) Rank B + Rank m For a connctd graph w hav 6. onfrnc Bishop Hbr ollg Thiruchirappalli 2 G. Nirmala and S. Priyadharshini P- Fuzzy Rank B = m-n+ algbra with Databas to facilitat uncrtainity Rank m Rank B Managmnt Aryabhatta Intrnational journal of Rank m ( m-n+) Mathmatics and informatics 22. Rank n- 7. G. Nirmala and D.R. Kirubaharan Optimal. ( 2 ) Matching procss for th manufactur of slind From () & (2) shaft by using ntwork thortical approach Rank of (G) = n- Aryabhatta, intrnational journal of mathmatics Hnc th thorm and informatics ISSN May 22. Rsult : If G is a Ptrson graph thn (G) is a dg cutst matrix of th ptrson graph with vrtics and 5 dgs Rank of (G) = n- Rank of (G) = 9 By thorm () onclusion Ptrson graph is a spcial kind of graph. ut st matrix is usd to communication and transportation ntwork Problms. In addition using th rlation. Of incidnc matrix, cycl matrix and dg cutst matrix and Ptrson graph cut sts matrix rank can b found. ut st 8. G. Nirmala and D.R Kirubaharan Ral lif problm on mad with RTS, Intrnational journal of scintific and Rsarch Publications, Vol-, May G. Nirmala and D.R. Kirubaharan Applications of Ntwork on ral lif problm applid Scinc priodical, vol.5, no-, Aug 2. Narasingh Do, Graph thory with applications to Enginring and computr Scinc.. S. Arumugam and S. Ramachandran Invitation to graph thory 2. Douglas B. Wst, Introduction to Graph thory. Michal Arting Algbra 4. John B.Fraligh A first cours in abstract algbra 25