PAijpam.eu SOME NEW SUM PERFECT SQUARE GRAPHS S.G. Sonchhatra 1, G.V. Ghodasara 2

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1 Internatonal Journal of Pure and Appled Mathematcs Volume 113 No , ISSN: (prnted verson); ISSN: (on-lne verson) url: do: /jpam.v PAjpam.eu SOME NEW SUM PERFECT SQUARE GRAPHS S.G. Sonchhatra 1, G.V. Ghodasara 2 1 School of Scence R.K. Unversty Rajkot, INDIA 1 Government Engneerng College Rajkot, Gujarat, INDIA 2 H. H.B. Kotak Insttute of Scence Rajkot, Gujarat, INDIA Abstract: A (p,q) graph G = (V,E) s called sum perfect square f for a bjecton f : V(G) {0,1,2,...,p 1} there exsts an njecton f : E(G) N defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(G). Here f s called sum perfect square labelng of G. In ths paper we derve several new sum perfect square graphs. AMS Subject Classfcaton: 05C78. Key Words: Sum perfect square graph, half wheel graph 1. Introducton We consder smple, fnte, undrected graph G = (p,q) (wth p vertces and q edges). The vertex set and the edge set of G are denoted by V(G) and E(G) respectvely. For all other termnology and notatons we follow Harary[1]. Sonchhatra and Ghodasara[4] ntated the study of sum perfect square graphs. Due to [4] t becomes possble to construct a graph, whose all edges can be labeled by dfferent perfect square ntegers. In [4] the authors proved that P n, C n, C n wth one chord, C n wth twn chords, tree, K 1,n, T m,n are sum Receved: December 28, 2016 Revsed: February 20, 2017 Publshed: March 28, 2017 c 2017 Academc Publcatons, Ltd. url: Correspondence author

2 490 S.G. Sonchhatra, G.V. Ghodasara perfect square graphs. In ths paper we prove that half wheel, corona, mddle graph, total graph, K 1,n +K 1, K 2 +mk 1 are sum perfect square graphs. Defnton 1.1. Let G = (p,q) be a graph. A bjecton f : V(G) {0,1,2,..., p 1} s called sum perfect square labelng of G, f the nduced functon f : E(G) N defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v) s njectve, uv E(G). A graph whch admts sum perfect square labelng s called sum perfect square graph. Defnton 1.2. The corona product G H of two graphs G and H s obtaned by takng one copy of G and V(G) copes of H and by jonng each vertex of the th copy of H to the th vertex of G by an edge, 1 V(G). Defnton 1.3. The mddle graph of a graph G denoted by M(G) s the graph wth vertex set V(G) E(G), where two vertces are adjacent f and only f ether they are adjacent edges of G or one s a vertex of G and other s an edge ncdent wth t. Defnton 1.4. The total graph of a graph G denoted by T(G) s the graph wth vertex set s V(G) E(G) where two vertces are adjacent f and only f (1) x,y V(G) are adjacent. (2) x,y E(G) are adjacent. (3) x V(G), y E(G) and y s ncdent to x. Defnton 1.5. Half wheel graph, denoted by HW n s constructed by the followng steps. Step 1: Consder a star K 1,n. Let {v 1,v 2,...,v n } be the pendant vertces of K 1,n and v be the apex vertex of K 1,n. Step 2: Add an edge between v and v +1, 1 n 2. Note that V(H(W n )) = n+1, E(H(W n )) = n+ n 2. Defnton 1.6. Let G 1 and G 2 be two graphs such that V(G 1 ) V(G 2 ) = φ. The jon of G 1 and G 2, denoted by G 1 +G 2, s the graph wth V(G 1 +G 2 ) = V(G 1 ) V(G 2 )ande(g 1 +G 2 ) = E(G 1 ) E(G 2 ) J, wherej = {uv/u V(G 1 ) and v V(G 2 )}.

3 SOME NEW SUM PERFECT SQUARE GRAPHS Man Results Observaton 1: If G = (V,E) s not sum perfect square graph, then ts supergraph s also not sum perfect square graph, but the converse may not be true. Fgure 1: A non sum perfect square graph K 4 wth sum perfect square subgraph K 4 {e}. In [4] Sonchhatra and Ghodasara posed the followng conjecture. Conjecture 2.1. AnoddsmplegraphGwth δ(g) = 3s not sumperfect square. Here we prove ths conjecture by usng the prncple of mathematcal nducton on number of vertces of the graph. Theorem 2.2. An odd smple graph G wth δ(g) = 3 s not sum perfect square. Proof. For any graph G = (V,E) wth V(G) = n, snce d(v) 3, v G, n must be even and n 4. We prove ths conjecture by usng prncple of mathematcal nducton on number of vertces n of graph G. Step 1. For n = 4, G = K 4 s not sum perfect square graph (See [4]). Step 2. Suppose the result s true for n = k, 4 < k < n. Step 3. Let G = (V,E ) be the graph wth V = k +1. H = G {v} s the graph wth k vertces, where v s any arbtrary vertex of G. By nducton hypothess, H s not a sum perfect square graph. Snce G s a supergraph of H, t s not a sum perfect square graph (Observaton 1). Theorem 2.3. C n K 1 s sum perfect square graph, n 3. Proof. Let V(C n K 1 ) = {u ;1 n} {v ;1 n}, where u 1,u 2,...,u n are successve vertces of C n and v 1,v 2,...,v n be the successve

4 492 S.G. Sonchhatra, G.V. Ghodasara vertces correspondng to n copes of K 1, E(C n K 1 ) = {e (1) = u u +1 ;1 n 2} {e (1) n 1 = u nu 1 } {e (2) = u v ;1 n}. We note that V(C n K 1 ) = 2n and E(C n K 1 ) = 2n. We defne a bjecton f : V(C n K 1 ) {0,1,2,...,2n 1} as f(u 1 ) = 0, { 4 6; 2 n f(u ) = n 4+4; n 2 +2 n. f(v ) = f(u )+1,1 n. Let f : E(C n K 1 ) N be the nduced edge labelng functon defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(C n K 1 ). Injectvty for edge labels: For 1 n 2 + 1, f (e (1) ) s ncreasng n terms of f (u u +1 ) < f (u +1 u +2 ) and for n n, f (e (1) ) s decreasng n terms of f (u u +1 ) > f (u +1 u +2 ). Smlarly f (e (2) ) s ncreasng for 1 n 2 +1 and decreasng for n 2 +2 n. Clam: {f (e (1) ),1 n 2 + 1} {f (e (1) ), n n 1} f (e (1) n ) {f (e (2) ),1 n 2 +1} {f (e (2) +2 n}. We have Further It s clear that f (e (1) ) = ), n 2 { (8 8) 2 ;2 n 2. (8n 8+4) 2 ; n 2 +2 n 1. f (e (1) n 2 +1) = (4n 6)2,f (e (1) 1 ) = 2,f (e (1) n ) = 4. f (e (2) ) = { (8 11) 2 ;2 n (8n 8+9) 2 ; n 2 +2 n. (1) f (e (2) 1 ) = 1, f (e (1) 1 ) = 2 and f (e (1) n ) = 4 are three smallest edge labels among all the edge labels n ths graph. (2) {f (e (1) ),1 n} are even, {f (e (2) ),1 n} are odd.

5 SOME NEW SUM PERFECT SQUARE GRAPHS 493 (3) f (e n 2 +1 ) s larger than the largest edge label of {f (e (1) ),1 n 2 } and smaller than the smallest edge label of {f (e (1) ), n 2 +2 n }. Hence we only need to prove the followng. (1) {f (e (1) 1 );2 n 2 } {f (e (1) ), n 2 +2 n 1}. (2) {f (e (2) 1 );1 n 2 } {f (e (2) ), n 2 +2 n}. Assume f possble {f (e (1) 1 );2 n 2 } = {f (e (1) ), n 2 +2 n 1}, for some. = 8 8 = 8n 8+4 or 8 8 = 8 8n 4. = = 2n+3 4 or n = 1 2, whch contradcts wth the choce of and n as,n N. Assume f possble {f (e (2) 1 );1 n 2 } {f (e (2) ), n n}, for some. = 8 11 = 8n 8+9 or 8 11 = 8 8n 9. = = 2n+5 4 or n = 1 4, whch contradcts wth the choce of as N. Hence f : E(C n K 1 ) N s njectve. So C n K 1 s sum perfect square graph, n 3. The below llustraton provdes better dea about the above defned labelng pattern. Fgure 2 : Sum perfect square labelng of C 6 K 1. Theorem 2.4. M(P n ) s sum perfect square graph, n N. Proof. Let V(M(P n )) = {v ;1 n} {v ;1 n 1}, where v s adjacent wth v +1, v and v +1, for 1 n 1. Here E(M(P n )) = {e = v v +1 ;1 n 2} {e(1) = v v ;1 n 1} {e(2) = v +1 v ;1 n 1}. We note that V(M(P n )) = 2n 1 and E(M(P n )) = 3n 4. We defne the bjecton f : V(M(P n )) {0,1,2,...,2n 2} as f(v ) = 2 2, 1 n, f(v ) = 2 1, 1 n 1.

6 494 S.G. Sonchhatra, G.V. Ghodasara Let f : E(M(P n )) N be the nduced edge labelng functon defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(M(P n )). Injectvty for edge labels: For 1 n 2, f (e ) s ncreasng n terms of f (v v +1 ) < f (v +1 v +2 ), 1 n 3. Smlarly f (e (1) ) and f (e (2) ) are also ncreasng, 1 n 1. Clam: {f (e );1 n 2} {f (e (1) );1 n 1} {f (e (2) );1 n 1}. We have f (e ) = (4) 2, 1 n 2 and f (e (1) ) = (4 3) 2, f (e (2) ) = (4 1) 2, 1 n 1. f (e ) are even, 1 n 2 and f (e (j) ) are odd, 1 n 1, j = 1,2. Assume f possble f (e (1) ) = f (e (2) ), for some, 1 n 1. = 4 3 = 4 1 or 4 3 = 1 4 = 3 = 1 or = 1 2, whch contradcts the choce of, as N. So f : E(M(P n )) N s njectve. Hence M(P n ) s sum perfect square graph, n N. The below llustraton provdes the better dea of the above defned labelng pattern. Fgure 3 : Sum perfect square labelng of M(P 7 ). Theorem 2.5. T(P n ) s sum perfect square graph, n N. Proof. Let V(T(P n )) = {v ;1 n} {v ;1 n 1}, where v s adjacent wth v +1, v and v +1, 1 n 1. E(T(P n )) = {e = v v +1 ;1 n 2} {e(1) = v v ;1 n 1} {e (2) = v +1 v ;1 n 1} {e(3) = v v +1 ;1 n 1}. V(T(P n )) = 2n 1 and E(T(P n )) = 4n 5. We defne a bjecton f : V(T(P n )) {0,1,2,...,2n 2} as f(v ) = 2 2, 1 n, f(v ) = 2 1, 1 n 1. Let f : E(T(P n )) N be the nduced edge labelng functon defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(T(P n )).

7 SOME NEW SUM PERFECT SQUARE GRAPHS 495 Injectvty for edge labels: For 1 n 2, f (e ) s ncreasng n terms of f (v v +1 ) < f (v +1 v +2 ), 1 n 3. Smlarly f (e (j) ) are also ncreasng, 1 n 1, 1 j 3. Clam: {f (e );1 n 2} {f (e (1) );1 n 1} {f (e (2) );1 n 1} {f (e (3) );1 n 1}. f (e ) = (4) 2, 1 n 2. For 1 n 1, f (e (1) ) = (4 3) 2, f (e (2) ) = (4 1) 2, f (e (3) ) = (4 2) 2. {f (e ),1 n 2}, f (e (3) ) areeven andf (e (j) )areodd, for1 n 1, j = 1,2,3. It s enough to prove the followng. (1) {f (e ),1 n 2} {f (e (3) ),1 n 1}. (2) {f (e (1) ),1 n 1} {f (e (2) ),1 n 1}. Assume f possble {f (e ),1 n 2} = {f (e (3) ),1 n 1}, for some. = 4 = 4 2 or 4 = 2 4. = 1 = 2 or = 1 4, whch contradcts wth the choce of, as N. Assume f possble {f (e (2) ),1 n 1} = {f (e (3) ),1 n 1}, for some. = 4 3 = 4 1 or 4 3 = 1 4. = 3 = 1 or = 1 2, whch contradcts wth the choce of, as N. So f : E(T(P n )) N s njectve. Hence T(P n ) s sum perfect square graph, n N. The below llustraton provdes the better dea of the above defned labelng pattern. Fgure 4 : Sum perfect square labelng of T(P 5 ). Theorem 2.6. HW n s sum perfect square graph, n N.

8 496 S.G. Sonchhatra, G.V. Ghodasara Proof. Let V(HW n ) = {v} {v ;1 n} and E(HW n ) = {e = vv ;1 n} {e (1) = v v +1 ;1 n 2 }. V(HW n) = n + 1 and E(HW n ) = n+ n 2. We defne a bjecton f : V(HW n ) {0,1,2,...,n} as f(v) = n, f(v ) = 1, 1 n. Let f : E(HW n ) N be the nduced edge labelng functon defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(HW n ). Injectvty for edge labels: For 1 n, f (e ) s ncreasng n terms of f (vv ) < f (vv +1 ), 1 n 1. Smlarly f (e (1) ) s also ncreasng. The largest edge label of f (e (1) ) s smaller than the smallest edge label of f (e ), therefore {f (e );1 n} {f (e (1) );1 n 2 }. Hence the nduced edge labelng f : E(H(W n )) N s njectve. So HW n s sum perfect square graph, n N. The below llustraton gves the better understandng of above defned labelng pattern. Fgure 5 : Sum perfect square labelng of H(W 6 ) and H(W 7 ). Theorem 2.7. K 1,n +K 1 s sum perfect square graph, n N. Proof. LetV(K 1,n +K 1 ) = {v} {v ;1 n} {w}, where{v 1,v 2,...,v n } are the pendant vertces and v s the apex vertex of K 1,n and w s the apex vertex correspondng to K 1. Here E(K 1,n + K 1 ) = {e (1) = vv ;1 n} {e (2) = wv ;1 n} {e = vw}. Note that V(K 1,n + K 1 ) = n + 2 and E(K 1,n +K 1 ) = 2n+1. We defne a bjecton f : V(K 1,n +K 1 ) {0,1,2,...,n+1} as f(v) = 0, f(v ) =, 1 n, f(w) = n+1. Let f : E(K 1,n +K 1 ) N be the nduced edge labelng functon defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(K 1,n +K 1 ). Injectvty for edge labels: For 1 n, f (e (1) ) s ncreasng n terms of f (vv ) < f (vv +1 ), 1 n 1. Smlarly f (e (1) ) s also ncreasng,

9 SOME NEW SUM PERFECT SQUARE GRAPHS 497 for 1 n. Clam: {f (e (1) );1 n} {f (e (2) );1 n} f (e). We have f (e (1) ) = () 2, 1 n, f (e (2) ) = (n++1) 2, 1 n and f (e) = (n+1) 2. The largest edge label of f (e (1) ) s smaller than the smallest edge label of f (e (2) ). If {f (e (1) ),1 n} = {f (e)} for some, then = n+1 or = n 1, whch contradcts wth the choce of, as N. Further the smallest edge label of f (e (2) ) s larger than f (e). Therefore {f (e (2) );1 n} f (e). So f : E(K 1,n +K 1 ) N s njectve. Hence K 1,n +K 1 s sum perfect square graph. The below llustraton provdes the better dea of the above defned labelng pattern. Fgure 6 : Sum perfect square labelng of K 1,3 +K 1. Theorem 2.8. K 2 +mk 1 s sum perfect square graph, m N. Proof. Let V(K 2 +mk 1 ) = {u 1,u 2 } {v ;1 m}, where {u 1,u 2 } be the vetex set of K 2. E(K 2 +mk 1 ) = {e = u 1 u 2 } {e (1) = u 1 v ;1 m} {e (2) = u 2 v ;1 m}. Here V(K 2 +mk 1 ) = m+2 and E(K 2 +mk 1 ) = 2m+1. We defne the bjecton f : V(K 2 +mk 1 ) {0,1,2,...,m+1} as f(u 1 ) = 0, f(u 2 ) = m+1 and f(v ) =, 1 m. Let f : E(K 2 +mk 1 ) N be the nduced edge labelng functon defned by f (uv) = (f(u)) 2 +(f(v)) 2 +2f(u) f(v), uv E(K 2 +mk 1 ). Injectvty for edge labels: For 1 m, snce f(v ) s ncreasng n terms of f (u j v ) < f (u j v +1 ), j = 1,2, 1 m 1.

10 498 S.G. Sonchhatra, G.V. Ghodasara Clam: f (e) {f (e (1) );1 m} {f (e (2) );1 m}. We have f (e) = (m+1) 2, f (e (1) ) = () 2, f (e (2) ) = (m++1) 2, 1 m. The largest edge label of f (e (1) ) s smaller than the smallest edge label of f (e (2) ). Also f (e) s larger than the hghest edge label of f (e (1) ) and smaller than the smallest edge label of f (e (2) ). Hence the clam s proved. So the nduced edge labelng f : E(K 2 +mk 1 ) N s njectve. So K 2 +mk 1 s sum perfect square graph, m N. The below llustraton provdes the better dea of the above defned labelng pattern. Fgure 7 : Sum perfect square labelng of K 2 +3K Concluson In ths paper a conjecture related to sum perfect square graph have been proved, a new graph called half wheel have been presented and varous sum perfect square graphs are found.

11 SOME NEW SUM PERFECT SQUARE GRAPHS 499 References [1] F. Harary, Graph Theory, Addson-wesley, Readng, MA, (1969). [2] J.A. Gallan, A dynemc survey of graph labelng, The Electroncs Journal of Combnatorcs, 18 (2015) [3] J. Shama, Square sum labelng for some mddle and total graphs, Internatonal Journal of Computer Applcatons, 37 (2012) 6-8, do: / [4] S.G. Sonchhatra, G.V. Ghodasara, Sum perfect square labelng of graphs, Internatonal Journal of Scentfc and Innovatve Mathematcal Research, 4 (2016)

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