Some cycle and path related strongly -graphs

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1 Some cycle ad path related strogly -graphs I. I. Jadav, G. V. Ghodasara Research Scholar, R. K. Uiversity, Rajkot, Idia. H. & H. B. Kotak Istitute of Sciece,Rajkot, Idia. gaurag Abstact A graph with vertices is said to be strogly -graph if its vertices ca be assiged the values {,,..., } i such a way that whe a edge whose ed vertices are labeled i ad j, is labeled with the value i + j + ij such that all edges have distict labels. Here we derive differet strogly -graphs i cotext of some graph operatios. Key words : Strogly -labelig, Strogly graph. Subject classificatio umber: C. I. I NTRODUCTION By a graph G, we mea a simple, fiite, udirected graph. Defiitio I.. [] A graph G with vertices is said to be strogly -graph if there is a bijectio f : V (G) {,,..., } such that the iduced edge fuctio f : E(G) N defied as f (e uv) f (u) + f (v) + f (u) f (v) is ijective. Here f is called strogly -labelig of graph G. Defiitio I.. [] The cycle with triagle is a cycle with three chords which by themselves form a triagle. For positive itegers p, q, r ad with p + q + r +, C (p, q, r) deotes the cycle with triagle whose edges form the edges of cycles Cp+, Cq+ ad Cr+ without chords. J Defiitio I.. [] The crow (C K ) is obtaied by joiig a pedat vertex to each vertex of cycle C by a edge. Defiitio I.. [] Tadpole, T (l, r) is the graph i which path of legth r is attached to ay oe vertex of cycle Cl by a bridge. T (l, r) has l + r vertices ad l + r edges. Defiitio I.. [] A chord of cycle C,, is a edge joiig two o-adjacet vertices of C. Defiitio I.. [] The middle graph of a graph G (with two or more vertices), deoted by M (G), is the graph whose vertex set is V (G) E(G) i which two vertices are adjacet if ad oly if either they are adjacet edges of G or oe is a vertex of G ad the other is a edge icidet o it. Defiitio I.. [] Two chords of cycle C,, are said to be twi chords if they form a triagle with a edge of C. For positive itegers ad p with p ( ), C,p is the graph cosistig of a cycle C with twi chords where the chords form the cycles Cp, C ad C+ p without chords with the edges of C. Defiitio I.. [] The total graph of a graph G (with two or more vertices), deoted by T (G), is the graph whose vertex set is V (G) E(G) ad two vertices are adjacet wheever they are adjacet vertices or adjacet edges i G or oe is a vertex of G ad the other is a edge icidet o it. ISSN: - Page

2 Defiitio I.9. [] The split graph of a graph G, deoted by spl(g), is the graph obtaied by addig a ew vertex v to each vertex v such that v is adjacet to every vertex that is adjacet to v i G. Defiitio I.. []The shadow graph D (G) of a coected graph G is costructed by takig two copies of G ad G of graph G. Joi each vertex u of G to the eighbours of the correspodig vertex u of G. mootoically icreasig (cosecutive) eve umbers. Hece the produced edge labels must be differet. Further the chord has (, ) label which is uique with respect to labelig f of the graph. Hece above defied labelig patter satisfies the coditios of strogly -labelig. i.e. Cycle C with oe chord is strogly -graph. Example. Strogly -labelig of cycle C with oe chord is show i Figure. Adiga ad Somashekara[] proved that all trees, cycles ad grids are strogly graphs. I the same paper they cosidered the problem of determiig the maximum umber of edges i ay strogly -graph of give order ad relates it to the correspodig problem for strogly multiplicative graphs. For all others stadard termiology ad otatios we follow Harary[]. II. M AIN R ESULTS Theorem II.. Cycle C with oe chord is strogly -graph for all N, where chord forms a triagle with two edges of C. Proof. Let G be the cycle C with oe chord. Let {v, v,..., v } deote the successive vertices of C, where v is adjacet to v ad vi is adjacet to vi+, i. Let e v v be the chord of C. Note that d(v ) d(v ) ad d(vi ), i, i. We defie vertex labelig fuctio f : V (G) {,,..., } as follows. ( f (vi ) i ; i d e. ( i + ); (d e + ) i. Hece the vertex labels o oe half of C are mootoically icreasig (cosecutive) odd umbers, whereas vertex labels o the other half of C are ISSN: - Fig. Corollary. Cycle C with twi chords C, is strogly -graph for all N. Proof. Let {v, v,..., v } deote the successive vertices of the C, where v is adjacet to v ad vi is adjacet to vi+, i. Let e v v ad e v v be two chords of C,. Note that d(v ), d(v ) d(v ) ad d(vi ), i, i, i. We defie vertex labelig fuctio f : V (C, ) {,,..., } same as per the labelig defied i Theorem. Here vertex labels o oe half of the C, are mootoically icreasig (cosecutive) odd umbers, whereas vertex labels o the other half of the C, are mootoically icreasig (cosecutive) eve umbers. Hece the produced edge labels must be differet. Further the vertex with label is adjacet to the vertices with label, ad which are uique with respect to the labelig f of the give graph. Hece above defied labelig patter satisfies the Page

3 coditios of strogly -labelig. i.e. C, with twi chords is strogly -graph. with triagle is strogly -graph show i Figure. Example. Strogly -labelig of cycle C with twi chords is show i Figure. Fig. Theorem II.. The crow C strogly -graph for all N. Fig. Corollary. Cycle with triagle C (,, ) is strogly -graph for all N. Proof. Let G be the cycle with triagle C (,, ). Let {v, v,..., v } deote the successive vertices of the G such that v is adjacet to v ad vi is adjacet to vi+, i. Let e v v, e v v ad e v v be three chords of C. Note that d(v ) d(v ) d(v ) ad d(vi ), i, i, i. We defie vertex labelig fuctio f : V (G) {,,..., } same as per the labelig defied i Theorem. Here vertex labels o oe half of the C (,, ) are mootoically icreasig (cosecutive) odd umbers, whereas vertex labels o the other half of the C (,, ) are mootoically icreasig (cosecutive) eve umbers. Hece the produced edge labels must be differet. Further the chords have (, ), (, ) ad (, ) labels which are uique with respect to labelig f of the graph G. Hece above defied labelig patter satisfies the coditios of strogly -labelig. i.e. C with triagle is strogly -graph. Example. Strogly -labelig of cycle C ISSN: - J K is., v } Proof. Let {v, v,..., v, v, v,..j be the vertices of the crow C K, where {v, v,..., v } are the vertices correspodig to cycle C ad {v, v,..., v } are the pedat vertices. Here vi is adjacet to vi, i,,...,. To defiej vertex labelig fuctio f : V (C K ) {,,,..., } we cosider the followig cases. Case : is odd. f (vi ) i; i. f (vi ) i ; i. Whe is odd, the labels for the vertices vi are cosecutive odd umbers, whereas the labels for the vertices vi are cosecutive eve umbers. Therefore the labels produced for the edges vi vi are odd ad i icreasig order. Further the edges vi vi+, i are labeled with eve labels i icreasig order. Hece all edges will have differet labels. Case : is eve. f (vi ) ( i ; i. ( i + ); ( + ) i. f (vi ) ( i ; i. ( i) + ; ( + ) i. Whe is eve, for ay two edge labels produced i the graph, {(j ) + (j ) + (j ) (j )} Page

4 {(j ) + (j + ) + (j ) (j + )} { + ( ) + ( )} {j + (j ) + j (j )} {j + (j + ) + j (j + )} 9, j. Hece above defied labelig patter satisfies the coditios J of strogly labelig. i.e. Crow C K is strogly graph for all. ed vertices labeled by vertex labels (i, j) ad (s, t) respectively, (i j s t) we have (i+j+ij) (s+t+st). Hece above defied labelig patter satisfies the coditios of strogly -labelig. i.e. T (l, r) is strogly -graph. Example. Strogly -labelig of Tadpole T (, ) is show i Figure. Example. Strogly -labelig of crow J C K is show i Figure. Fig. 9 Fig. Theorem II.. Middle graph of cycle (C ) is strogly -graph for all N. Theorem II.. Tadpoles T (l, r) strogly -graph for all l, r N. are Proof. Let {v, v,..., vl } be the vertices of the cycle Cl ad {vl+, vl+,..., vl+r } be the vertices of path Pr attached to vertex vl of Cl i the T (l, r). Let e vl vl+ be the bridge joiig vertex vl of Cl ad vertex vl+ of the Pr. Note that V (T (l, r)) E(T (l, r)) l + r. To defie vertex labelig fuctio f : V (T (l, r)) {,,,..., l + r}, We cosider the followig cases. P Case : l + m i i, where m N {, }. f (vi ) i, i l + Pr. Case : l + m i i, where m N {, }. f (vl ) l +, f (vl+ ) l. f (vi ) i, i l + r, i l, l +. Sice the labelig f defied above is strictly icreasig, for ay two edges with ISSN: - Proof. Let {v, v,..., v, v, v,..., v } be the vertices of the graph M (C ), where {v, v,..., v } be the vertices correspodig to the cycle C ad {v, v,..., v } be the vertices correspodig to the edges of C. We defie vertex labelig fuctio f : V (M (C )) {,,,..., } as follows. f (vi ) ( i ; i d e. ( i + ); d e + i. f((vi ) i ; i b c. ( i) + ; b c + i. Here vertex labels i oe half of the M (C ) are cosecutive eve umbers i icreasig order, whereas the vertex labels i other half of the M (C ) are cosecutive odd umbers i icreasig order. Page 9

5 Whe is eve, for ay eve edge label produced i graph, {(k)+(k+)+(k) (k+)} {(k+ ) + ((k + )) + (k + ) ((k + ))} {((k + )) + (k) + ((k + )) (k)} {() + () + () ()} ad for ay odd edge label produced i the graph, {(k )+(k )+(k ) (k )} {(k )+(k +)+(k ) (k +)} {(k +)+(k )+(k +) (k )} {( )+( )+( ) ( )} {( ) + () + ( ) ()} {( ) + () + ( ) ()} {()+()+() ()} {()+()+() ()}. where k. Whe is odd, for ay eve edge label produced i the graph {((k)+(k+)+(k) (k+)} {(k+ ) + ((k + )) + (k + ) ((k + ))} ((k + )) + (k) + ((k + )) (k)} {( ) + () + ( ) ()} {() + () + () ()}, where k b c ad for ay odd edge label produced i graph {(k )+(k )+(k ) (k )} {(k )+(k +)+(k ) (k +)} {(k +)+(k )+(k +) (k )} {( )+( )+( ) ( )} {( ) + () + ( ) ()} {()+()+() ()} {()+()+() ()}, where k b c. Hece above defied labelig patter satisfies the coditios of strogly labelig. i.e. M (C ) is strogly graph. Fig. vertices correspodig to path } be the P ad {v, v,..., v vertices correspodig to the edges {e, e,..., e } of P. We defie vertex labelig f : V (M (P )) {,,,..., } as follows. f (vi ) i, i. f (vi ) i, i. The labels for the vertices vi are cosecutive odd umbers, whereas the labels for the vertices vi are cosecutive eve umbers. Therefore the labels produced for the edges vi vi are odd ad i icreasig order. Further the edges vi vi+ ad ( i ) are labeled with eve label i icreasig order. Hece all edges will have differet label. So, labelig patter defied above satisfies the coditios of strogly -labelig. i.e. M (P ) is strogly -graph. Example. Strogly -labelig of M (P ) is show i Figure. Example. Strogly -labelig of M (C ) is show i Figure. Theorem II.. Middle graph of path P is strogly -graph for all N. Fig. Proof. Let {v, v,..., v, v, v,..., v } be the vertices of the M (P ), where {v, v,..., v } be the ISSN: - Page

6 Theorem II.. The total graph of P is strogly -graph for all N. Proof. Let } {v, v,..., v, v, v,..., v be the vertices of T (P ), where {v, v,..., v } be the vertices of } be the path P ad {v, v,..., v vertices correspodig to the edges {e, e,..., e } of P. We defie vertex labelig fuctio f : V (T (P )) {,,,..., } as follows. f (vi ) i, i. f (vi ) i, i. The labels for the vertices vi are cosecutive odd umbers, whereas the labels for the vertices vi are cosecutive eve umbers. Therefore the labels produced for the edges vi vi ad vi vi+ are odd ad i icreasig order. For the label of ay of the remarkig edges with commo ed vertices, we have {(i ) + (i + ) + (i ) (i + )} {(i) + (i + ) + (i) (i + )} {(i) + (i ) + (i) (i )}, i. Further the edges icidet with vi ad vi+ ( i ) are labeled with eve labels i icreasig order. Hece all edges will have differet labels. So, labelig patter defied above satisfies the coditios of strogly -labelig. i.e. M (P ) is strogly -graph. Example. Strogly -labelig of T (P ) is show i Figure as a illustratio for Theorem. Fig. Theorem II.. The split graph of P is ISSN: - strogly -graph for all N. Proof. Let {u, u,..., u, v, v,..., v } be the vertices of the spl(p ), where {v, v,..., v } are the vertices of the path P ad {u, u,..., u } are ewly added vertices correspodig to the vertices of P to obtai spl(p ). We defie vertex labelig fuctio f : V (spl(p )) {,,,..., } as follows. f (vi ) i, i. f (ui ) i, i. Here for the label of ay two edges with commo vertex, we have {(i ) + (i + ) + (i ) (i + )} {(i ) + (i + ) + (i ) (i + )} {i + (i + ) + i (i + )}. Therefor above defied labelig patter satisfies the coditios of strogly -graph. i.e. spl(p ) is strogly graph. Example 9. Strogly -labelig of spl(p ) is show i Figure 9 as a illustratio for Theorem. 9 Fig. 9 Theorem II.. The shadow graph D (P ) is strogly -graph for all N. Proof. Let {v, v,..., v, v, v,..., v } be the verties of the D (P ), where {v, v,..., v } are the vertices of path P ad {v, v,..., v } are the vertices added correspodig to the vertices {v, v,..., v } i order to obtai D (P ). We defie vertex labelig f : V (D (P )) {,,,..., } as follows. f (vi ) i, i. f (vi ) i, i. Page

7 The labels for the vertices vi are cosecutive odd umbers, whereas the labels for the vertices vi are cosecutive eve umbers. Therefore the labels produced for the edges vi vi+ are eve labels i icreasig order the labels for edges vi vi+, vi vi+ ad vi vi+ are odd labels i icreasig order. Also for ay two edge labels produce i the graph, {(i ) + (i + ) + (i ) (i + )} {(i ) + (i + ) + (i ) (i + )} {i + (i + ) + i (i + )} for ay i. So, the labelig patter defied above satisfies the coditios of strogly -graph. i.e. D (P ) is strogly -graph. Example. Strogly -labelig of D (P ) is show i Figure as a illustratio for Theorem. 9 Fig. REFERENCES [] J. A. Gallia, A dyemic survey of graph labelig, The Electroics Joural of Combiatorics, (), ]DS 9. [] F. Harary, Graph theory, Addiso-Wesley, Readig, Massachusetts, (99). [] C. Adiga ad D. Somashekara, Strogly -graphs, Math. Forum, (999), -. [] M. A. Seoud ad A. E. A. Mahra, Some otes o strogly -graphs, preprit. [] M. A. Seoud ad A. E. A. Mahra, Necessary coditios for strogly -graphs, AKCE It. J. Graphs Comb., 9, No. (),. ISSN: - Page