Article On the Additively Weighted Harary Index of Some Composite Graphs

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1 mathematics Article On the Aitively Weighte Harary Inex of Some Composite Graphs Behrooz Khosravi * an Elnaz Ramezani Department of Pure Mathematics, Faculty of Mathematics an Computer Science, Amirkabir University of Technology Tehran Polytechnic), 44, Hafez Ave, Tehran 594, Iran; elramezani@yahoocom * Corresponence: khosravibbb@yahoocom; Tel: Acaemic Eitor: Michael Falk Receive: 3 November 06; Accepte: February 07; Publishe: 7 March 07 Abstract: The Harary inex is efine as the sum of reciprocals of istances between all pairs of vertices of a connecte graph The aitively weighte Harary inex H A G) is a moification of the Harary inex in which the contributions of vertex pairs are weighte by the sum of their egrees This new invariant was introuce in Alizaeh, Iranmanesh an Došlić Aitively weighte Harary inex of some composite graphs, Discrete Math, 03) an they pose the following question: What is the behavior of H A G) when G is a composite graph resulting for example by: splice, link, corona an roote prouct? We investigate the aitively weighte Harary inex for these stanar graph proucts Then we obtain lower an upper bouns for some of them Keywors: aitively weighte harary inex; composite graph; corona; roote prouct; splice; link Introuction A topological inex is a real number erive from the structure of a graph in a way that oes not epen on the labeling of the vertices Hence, isomorphic graphs have the same values of topological inices Chemical graph theory is a branch of mathematical chemistry that is mostly concerne with fining topological inices of chemical graphs that correlate well with certain physico-chemical properties of the corresponing molecules The basic iea behin this approach is that the physico-chemical properties are governe by the mechanism epening mostly on the valences of atoms an on their relative positions within the molecule Since both concepts are well escribe in graph-theoretical terms, there are reasons to believe that chemical graphs capture enough information about real molecules to make them useful as their moels Hunres of ifferent topological inices have been investigate so far an have been employe in QSAR Quantitative Structure Activity Relationship)/QSPR Quantitative Structure Property Relationship) stuies, with various egrees of success Most of the more useful invariants belong to one of two broa classes: they are either istance base, or bon aitive The first class contains the inices that are efine in terms of istances between pairs of vertices; the secon class contains the inices efine as the sums of contributions over all eges Typical representants of the first type are the Wiener inex an its various moifications; characteristic for the secon type are the Ranić inex [] an the two Zagreb inices Another istance-base topological inex of the graph G is the Harary inex The Harary inex of a graph G, enote by HG), was introuce inepenently by Plavšić et al [] an by Ivanciuc et al [3] in 993 The Harary inex is efine as follows: HG) = {u,v} VG) G u, v), Mathematics 07, 5, 6; oi:03390/math50006 wwwmpicom/journal/mathematics

2 Mathematics 07, 5, 6 of 3 where the summation goes over all pairs of vertices of G an G u, v) enotes the istance of the two vertices u an v in the graph G For a list of new results about the Harary inex see [4 7] The aitively weighte version of the Harary inex was introuce by Alizaeh et al [8] in 03 For a given graph G, its aitively weighte Harary inex H A G) is efine as: H A G) = {u,v} VG) G u) + G v), G u, v) where G u) enotes the egree of vertex u in G It is obvious that, if G is a k-regular graph, then H A G) = khg) Also, in the paper [8], they pose the following question: What is the behavior of H A G) when G is a composite graph resulting for example by: splice, link, corona an roote prouct? In this paper we investigate the behavior of H A G) uner these four operations which are useful in chemistry Also, we try to obtain upper an lower bouns for H A G) of these operations Preliminary Results All graphs consiere in this paper are finite, simple an connecte For a given graph G we enote by VG) its vertex set, an by EG) its ege set The carinalities of these two sets are enote by n an e, respectively The egree of a vertex u VG) is enote by G u) an the istance G u, v) between vertices u an v in G is the length of any shortest path in G connecting u an v The iameter of the graph G, enote by DG), is max{ G u, v) u, v VG)} We enote by K n an P n the complete graph an the path graph with n vertices, respectively A regular graph is a graph where each vertex has the same number of neighbors A regular graph with vertices of egree k is calle a k-regular graph or regular graph of egree k The first an the secon Zagreb inices of a graph G are efine as follows: M G) = G u) + G v)), M G) = G u) G v) uv EG) uv EG) These topological inices were conceive in the 970s [9,0] In 008, in [] the first an the secon Zagreb coinices of a graph G are efine as follows: M G) = G u) + G v)), M G) = G u) G v) uv/ EG) uv/ EG) Also, the first an the secon Zagreb coinices of graph G with n vertices an e eges are equal to M G) = en ) M G) an M G) = e M G) M G), respectively For the proof of these facts, we refer the reaers to [] We will use Zagreb inices an Zagreb coinices to formulate our results in a more compact way For a graph G with u VG), we efine PG) = u,v VG) G u,v)+ an P Gv) = u VG) G u,v)+ Also we efine H Gv) = u VG)\{v} G u,v) In the rest of the paper any sum {u,v} VG) hu, v) enotes the sum u VG) hu, u) + {u,v} VG) hu, v), where hu, v) is the contribution of pair u, v to the sum In the sequel of this paper we enote by n G, e G for the number of vertices an the number of eges of G an we enote by n H, e H for the same quantities for H 3 Main Results In this section we introuce the stanar graph proucts resulting in composite graphs an then we present explicit formulas for the values of aitively weighte Harary inices of them

3 Mathematics 07, 5, 6 3 of 3 3 Roote Prouct Definition The roote prouct G{H} is obtaine by taking one copy of G an VG) copies of a roote graph H, an by ientifying the root of the i-th copy of H with the i-th vertex of G, i =,,, VG) For the roote prouct G{H} we have: VG{H}) = VG) VH), EG{H}) = EG) + VG) EH) As an example for roote prouct see Figure ON ADDITIVELY WEIGHTED HARARY INDEX OF SOME COMPOSITE GRAPH G G{H} : H w Figure The Figure roote The roote prouct of Gof ang H where anw is Hthewhere root of H w is the root of H Lemma Let G be a simple graph an H be a roote graph with w as its root Then for a vertex u of G{H} such that u VG), we have G{H} u) = G u) + H w), an G u,v)+ u V G) Also we efine H G u,v)+ Gv) = for a vertex v of G{H} such that v / VG) u V G)\{v} ) if v VG), then G{H} u, v) = G u, v), G u,v) ) if u VG), v VH i ), where i =,,, VG), then G{H} u, v) = G u, w i ) + Hi w i, v) = For a graph G with u V G), we efine P G) = u,v V G) we have G{H} v) = H v 0 ), where v 0 is the corresponing vertex in H as v of H i Also: In the rest of the paper any sum {u,v} V G) hu, v) enotes the sum u V G) G u, w i ) + H w, v 0 ), where w i is the root of H i an v 0 is the corresponing vertex in H as v of H i, 3) if u, v VH i ), where i =,,, VG), then G{H} u, v) = H u 0, v 0 ), where u 0 an v 0 are the {u,v} V corresponing G) hu, v), vertices where in H as uhu, an v of v) H i, is the contribution of pair u, v to the sum u v 4) if u VH i ), v VH j ) an i < j VG), then G{H} u, v) = Hi u, w i ) + Hj v, w j ) + G w i, w j ) = H u 0, w) + H v 0, w) + G w i, w j ), where w i is the root of H i an w j is the root of H j Also, u 0 an v 0 are the corresponing vertices in H as u of H i an v of H j, respectively In the sequel of this paper we enote by n G, e G for the number of vertice number of eges of G an we enote by n H, e H for the same quantities for H Proof The proof is straightforwar 3 Main results Theorem Let G be a simple graph an H be a roote graph with w as its root Then: In this sectionh A we G{H}) introuce = H A G) + the H w)hg) stanar + n G H A H) graph + e G Hproucts H w) resulting in composi an then we present explicit + formulas for the G u) + values H v) + of H w) aitively weighte Hara {u,t} VG) v VH)\{w} G u, t) + H v, w) of them u =t 3 Roote Prouct + {t,l} VG) t =l Proof From the efinition we have: Definition 3 The roote prouct G{H} is obtaine by taking one copy V G) copies of a roote graph H, an G{H} u) + by ientifying G{H} v) H A G{H}) = the root of the i th c {u,v} VG{H}) with the i th vertex of G, i =,,, V G) G{H} u, v) For the roote prouct G{H} we have {u,v} VH)\{w} H u) + H v) H u, w) + H v, w) + G t, l) an V G{H}) = V G) V H), EG{H}) = EG) + V G) EH)

4 Mathematics 07, 5, 6 4 of 3 By Lemma, we partition the sum into four sums S i, i =,, 3, 4, where: Hence: S = S = S 3 = S 4 = G{H} u) + G{H} v) {u,v} VG) G{H} u, v) = H A G) + H w)hg), n G i= u VG) v VH i )\{w i } n G i= {u,v} VH i )\{w i } i<j n G u VH i )\{w i } v VH j )\{w j } = {t,l} VG) t =l G{H} u) + G{H} v) {u,v} VH)\{w} G{H} u, v) G{H} u) + G{H} v) G{H} u, v) G{H} u) + G{H} v) G{H} u, v) H A G{H}) = H A G) + H w)hg) + + u VG) v VH)\{w} + {t,l} VG) t =l = G u) + G v) + H w) {u,v} VG) G u, v) = {u,t} VG) v VH)\{w} = n G {u,v} VH)\{w} H u) + H v) H u, w) + H v, w) + G t, l) G u) H v, w) + n G {u,v} VH)\{w} Thus we complete the proof of this theorem v VH)\{w} {w,v} VH) w =v {u,t} VG) u =t G u) + H w) + H v), G u, t) + H v, w) H u) + H v), H u, v) G u) + H w) + H v) G u, t) + H v, w) H w) + H v) H v, w) H u) + H v) H u, w) + H v, w) + G t, l) Example We have: H A P {K 3 }) = 48 3, H AP 3 {K 3 }) = 33 3 Base on Theorem, we obtain the next corollary immeiately Corollary Let G be a r-regular graph an H be a k-regular roote graph with w as its root Then: H A G{H}) = r + k)hg) + kn G HH) + n G rh H w) + r + k) + k {t,l} VG) t =l {u,t} VG) u =t v VH)\{w} {u,v} VH)\{w} G u, t) + H v, w) H u, w) + H v, w) + G t, l) We can etermine a lower an an upper boun for H A G{H}), where G is a r-regular graph an H is a k-regular roote graph

5 Mathematics 07, 5, 6 5 of 3 We know that G u, v) DG), where {u, v} VG), u = v an DG) is the iameter of G Similarly, we have H u, v) DH), where {u, v} VH), u = v an DH) is the iameter of H Hence, we have: 3 Corona H A G{H}) r + k)hg) + kn G HH) + n G rh H w) r + k + n G n G )n H )[ DH) + DG) + kn H ) DH) + DG) ], H A G{H}) r + k)hg) + kn G HH) + n G rh H w) + n G n G )n H )[ r + k + kn H ) ] 3 Definition Let G an H be two graphs The corona prouct G H is obtaine by taking one copy of G an VG) copies of H; an by joining each vertex of the i-th copy of H to the i-th vertex of G, i =,,, VG) For the corona prouct G H, we have: VG H) = VG) + VH) ), EG H) = EG) + VG) VH) + EH) ) As an example for the corona prouct see Figure 6 BEHROOZ KHOSRAVI & ELNAZ RAMEZANI G G H: H Figure Figure The The corona prouct of G an anh H Lemma Let G an H be two simple connecte graphs For a vertex u of G H such that u VG), For the corona prouct G H, we have we have G H u) = G u) + VH), an for a vertex v of G H such that v VH), we have G H V G v) = H) H = v) V + G) Also: + V H) ), EG H) = EG) + V G) V H) + EH) ) ) As anif u, example v VG), for then corona G H u, prouct v) = G u, seev), Figure ) if u VG), v VH i ), where i =,,, VG), then G H u, v) = G u, w i ) +, where w i is the Lemma 37 Let G an H be two simple connecte graphs For a vertex u of G H such i-th vertex in G, 3) that u if u, v V G), VH we have G H u) = G u) + V H), an for a vertex v of G H such that i ), where i =,,, VG), then: v V H), we have G H v) = H v) + Also { ) if u, v V G), then G H u, v) = G u, v), if uv EH ) if u V G), v V H i ), where G H u, i = v), = i ),, V G), if uv then EH G H i )u, v) = G u, w i )+, where w i is the i th vertex in G, 4) 3) if uif u, VH v i ), V vh i VH ), where j ) ani =, i, <, j V VG), then G H u, v) = G w i, w j ) +, where w i is the i-th an w { j is the j-th vertices in G if uv EHi ) G H u, v) = if uv EH i ) Proof The proof is obvious 4) if u V H i ), v V H j ) an i < j V G), then G H u, v) = G w i, w j )+, Lemmawhere 3 Let Gw i beisa the simple i graph th an anw K j is be the complete j th vertices graph ofin orer G Then: Proof The proof is obvious HG{K }) = HG) + PG) + Lemma 38 Let G be a simple graph an K be the complete {u,v} VG) G graph u, v) + of orer Then HG{K }) = HG) + P G) + G u, v) + Proof By efinition {u,v} V G) u v

6 Mathematics 07, 5, 6 6 of 3 Proof By efinition: HG{K }) = {u,v} VG{K }) G{K }u, v) We partition the sum in the formula of HG{K }) into three sums S i such that S i is over A i for i =,, 3, where: A = {u, v) u, v VG)}, A = {u, v) u VG), v V K ) i ) \ {wi }, i VG) }, A 3 = {u, v) u V K ) i ) \ {wi }, v V K ) j ) \ {wj }, i < j VG) }, where K ) i is the i-th copy of K an K ) j is the j-th copy of K in G{K } So we have: HG{K }) = S + S + S 3 ng = {u,v} VG) G{K } u, v) + i= + u VG) v VK ) i )\{w i } i<j n G u VK ) i )\{w i } G{K }u, v) v VK ) j )\{w j } ng = {u,v} VG) G u, v) + i= = HG) + PG) + {u,v} VG) G u, v) + G{K }u, v) u VG) G u, w i ) + + i<j n G G w i, w j ) + Theorem Let G an H be simple graphs Then: H A G H) = H A G) + n H e H n H )HG) + n H e H + n H )HG{K }) Proof By efinition we have: + n G M H) + [e H n H ) n H + n H]PG) + n G n H e H + n H ) + n H {u,v} VG) H A G H) = {u,v} VG H) G u) G u, v) + G H u) + G H v) G H u, v) By Lemma, we partition the sum into four sums S i, i =,, 3, 4 We consier four sums S, S, S 3, S 4 as follows: S = S = G H u) + G H v) {u,v} VG) G H u, v) = H A G) + n H HG) VG) i= u VG) v VH i ) G H u) + G H v) G H u, v) = G u) + G v) + VH) {u,v} VG) G u, v) = n G i= u VG) v VH i ) G u) + VH) + H v) + G u, w i ) +

7 Mathematics 07, 5, 6 7 of 3 Now we consier the following relation: G u) + H v) + n H + n = H u VG) G u, w i ) + G u) + e H u VG) G u, w i ) + + n H + )n H u VG) v VH i ) Hence, we have: S = S 3 = n G [n H i= u VG) = n H u VG) G u) G u, w i ) + + e H + n H + n H) G u, w i ) + u VG) = n H G u) u VG) G u, w i ) + + e H + n H + n H)P G w i ) G u) G u, w i ) + + e H + n H + n H)P G w i )] = n H G u) {u,v} VG) G u, v) + + e H + n H + n H)PG) VG) i= {u,v} VH i ) G H u) + G H v) G H u, v) Now we consier the following relation: = n G i= {u,v} VH i ) H u) + H v) + G H u, v) G u, w i ) + H u) + H v) + = {u,v} VH) G H u, v) H u) + H v) + ) + H u) + H v) + uv EH) uv/ EH) = M H) + ) M VH) H) + EH) + [ EH) ] = M H) + n H e H + n H ) Note that the last equality hols in view of the fact that M H) = e H n H ) M H) So: S 3 = S 4 = n G i= [ M H) + n H e H + n H i<j n G u VH i ) v VH j ) G H u) + G H v) G H u, v) )] = n G [ M H) + n H e H + n H )] = i<j n G u VH i ) v VH j ) H u) + H v) + G w i, w j ) + Now we consier the following relation, where i < j n G H u) + H v) + = u VH i ) G w i, w j ) + v VH j ) By using Lemma 3 we have: S 4 = n H e H + n H ) {u,v} VG) G w i, w j ) + [ u VH i ) v VH j ) H u) + H v)) + n H ] = n He H + n H ) G w i, w j ) + G u, v) + = n He H + n H )[HG{K }) HG) PG)] The result now follows by aing the four contributions an simplifying the expression

8 Theorem 3 Let G an H be two simple graphs For vertices y VG) an z consier G H)y; z) Then Mathematics 07, 5, 6 8 of 3 Example H A P K ) = 48 3, H A P 3 K ) = 33 3 We note that P K = P {K 3 } an P 3 K = P 3 {K 3 } an the result is similar to Example Corollary Let G be a r-regular graph an H be a simple graph Then: H A G H) = [n H e H n H ) + r]hg) + [e H n H ) n H + n Hr + )]PG) + n G M H) + n H e H + n H )HG{K }) + n G n H e H + n H ) 33 Splice Definition 3 For given vertices y VG) an z VH) the splice of G an H by vertices y an z, which is enote by G H)y; z), is efine by ientifying the vertices y an z in the union of G an H Mathematics 07, 5, x As an example for the splice of G an H see Figure 3 G y z G H)y; z): H Figure 3 The Figure splice 3 The splice of Gof G an H by vertices y an z y an z Then for the splice of G an H by vertices y an z we have: Then for the splice of G an H by vertices y an z we have V G H)y; z) ) = VG) + VH), E G H)y; z) ) = EG) + EH) Lemma 4 Let G an H be simple graphs with isjoint vertex sets For given vertices y VG) an z VH) suppose that the splice of G an H by vertices y an z is enote by G H for convenience Then for a vertex u of G H such that u VG) \ {y} we have G H u) = G u) an for a vertex v of G H such that v VH) \ {z} we have G H v) = H v) an G H y) = G y) + H z) = G H z) Also: V G H)y; z) ) = VG) + VH), E G H)y; z) ) = EG) + EH) Lemma 4 Let G an H be simple graphs with isjoint vertex sets For given vertices y VG) an suppose ) ifthat u, v the VG), splice then of G H Gu, an v) = H G by u, v), vertices y an z is enote by G H for convenience Then ) if u, v VH), then u of G G H u, v) = H such that u VG) \ H u, v), 3) if u VG), v VH), then G H u, v) {y} = we G u, have y) + H z, G H v) u) = G u) an for a vertex v of G H v VH) \ {z} we have G H v) = H v) an G H y) = G y) + H z) = G H z) Also Proof The proof is obvious if u, v VG), then G H u, v) = G u, v), Theorem 3 Let G an H be two simple graphs For vertices y VG) an z VH), consier G H)y; z) Then: if u, v VH), then G H u, v) = H u, v), 3 if u VG), v VH), then G H u, v) = G u, y) + H z, v) Proof The proof is obvious H A G H)y; z) ) = HA G) + H A H) + H z)h G y) + G y)h H z) + u VG)\{y} G u) + H v) G u, y) + H v, z)

9 Mathematics 07, 5, 6 9 of 3 Proof For convenience we enote G H)y; z) by G H By efinition we have: H A G H) = {u,v} VG H) G H u) + G H v) G H u, v) We partition the sum into three sums S i such that S i is over A i for i =,, 3, where A = {u, v) u, v VG)}, A = {u, v) u, v VH)}, A 3 = {u, v) u VG) \ {y}, v VH) \ {z}} So we have: S = G H u) + G H v) {u,v} VG) G H u, v) = G u) + G v) + {u,v} VG)\{y} G u, v) G y) + H z) + G v) v VG)\{y} G y, v) = H A G) + H z)h G y) Similarly, we have S = H A H) + G y)h H z) Also: S 3 = G H u) + G H v) = u VG)\{y} G H u, v) u VG)\{y} The result now follows by aing the three sums S i, i =,, 3 G u) + H v) G u, y) + H v, z) Corollary 3 Let G be a r-regular graph an H be a k-regular graph For vertices y VG) an z VH), consier G H)y; z) Then: H A G H)y; z) ) = rhg) + khh) + khg y) + rh H z) + r + k) u VG)\{y} G u, y) + H v, z) We can etermine a lower an an upper boun for H A G H)y; z) ), where G an H are r-regular an k-regular graphs, respectively We know that G u, y) DG), where u VG) \ {y} an DG) is the iameter of G Similarly, we have H v, z) DH), where v VH) \ {z} an DH) is the iameter of H Hence, we have: ) H A G H)y; z) rhg) + khh) + khg y) + rh H z) + r + k)n G )n H ), DG) + DH) ) H A G H)y; z) rhg) + khh) + khg y) + rh H z) + r + k)n G )n H ) 34 Link Definition 4 A link of G an H by vertices y an z, which is enote by G H)y; z), is efine as the graph obtaine by joining y an z by an ege in the union of these graphs As an example of the link of two graphs see Figure 4

10 Definition 36 A link of G an H by vertices y an z, which is enot H)y; z), is efine as the graph obtaine by joining y an z by an ege in Mathematics 07, 5, 6 0 of 3 these graphs G y H z G H)y; z): y z Figure 4 The Figure link4 of TheGlinkan of G an H by vertices vertices y an z y an z For a link of G an H by vertices y an z we have: V G H)y; z) ) = VG) + VH), E G H)y; z) ) = EG) + EH) + Lemma 5 Let G an H be two simple graphs with isjoint vertex sets For given vertices y VG) an zas VH) suppose a link of G an H by vertices y an z is enote by G an example of the link of two graphs H for convenience Then for a vertex see Figure 4 u of G H such that u VG) \ {y} we have G H u) = G u) an for a vertex v of G H such that vfor VH) a link \ {z} we of have G G H an v) H= H by v) an vertices G H y) = y G an y) + z, we G H z) have = H z) + Also: ) if u, v VG), then G H u, v) = G u, v), V G H)y; z) ) = V G) + V H), E G H)y; z) ) ) if u, v VH), then G H u, v) = H u, v), = EG) + 3) if u VG), v VH), then G H u, v) = G u, y) + H z, v) + Lemma Proof The proof 37 is straightforwar Let G an H be two simple graphs with isjoint vertex sets vertices Theorem y4 Let V G an G) H be an two simple z V graphs H) For suppose vertices y a VG) linkan ofz G VH), anconsier H bygvertices y an byh)y; G z) Then: H for convenience Then for a vertex u of G H such that u V have G H H A G u) = H)y; G z)) u) = H A an G) + for H A H) a + vertex H G y) + Hv H z) of G H such that v V H) \ G H v) = H v) an G H + H z) y) + )P = G G y) y) ) +, G y) G H + )P z) H z) = ) H z) + Also ) if u, v V G), then G H u, v) = G u, v), + G y) + H z) + + ) if u, v V H), then + G H u, v) = H u, v), u VG)\{y} G u) G u, y) + + 3) if u V G), v V H), then G H u, v) = G u, y) + H z, v) + Proof The proof is straightforwar {u,v} VG H) u VG)\{y} G u) + H v) G u, y) + H v, z) + H v) H v, z) + Proof For convenience we enote G H)y; z) by G H By efinition we have: H A G H) = G H u) + G H v) G H u, v) Similarly to the proof of Theorem 3, we partition the sum into three sums S i such that S i is over A i for i =,, 3, where: A = {u, v) u, v VG)}, A = {u, v) u, v VH)}, A 3 = {u, v) u VG), v VH)}

11 Mathematics 07, 5, 6 of 3 We consier three sums S, S, S 3 as follows: S = G H u) + G H v) {u,v} VG) G H u, v) = G u) + G v) + {u,v} VG)\{y} G u, v) G u) + G y) + u VG)\{y} G u, y) = H A G) + H G y) Similarly, we have S = H A H) + H H z) Also: S 3 = G H u) + G H v) u VG) G H u, v) v VH) = u VG)\{y} G u) + H v) G u, y) + H v, z) + + u VG)\{y} + Gy) + H v) + ) + H v, z) + G y) + H z) + = G u) + H v) u VG)\{y} G u, y) + H v, z) + + Gy) + H z) + + G u) u VG)\{y} G u, y) + + Hz) + )P G y) ) + H v) H v, z) + + Gy) + )P H z) ) G u) + H z) + G u, y) + We obtain the result by aing the three sums S i, i =,, 3 Corollary 4 Let G be a r-regular graph an H be a k-regular graph For vertices y VG) an z VH), consier G H)y; z) Then: H A G H)y; z) ) = rhg) + khh) + HG y) + H H z) + k + r + )[P G y) + P H z) ] + + k + r) u VG)\{y} G u, y) + H v, z) + Similarly, we can etermine a lower an an upper boun for H A G H)y; z) ), where G an H are r-regular an k-regular graphs, respectively We know that G u, y) DG), where u VG) \ {y} an DG) is the iameter of G Similarly, we have H v, z) DH), where v VH) \ {z} an DH) is the iameter of H So we have: H A G H)y; z) ) rhg) + khh) + HG y) + H H z) + r + k + )[P G y) + P H z) ] + + r + k)n G )n H ), DG) + DH) + H A G H)y; z) ) rhg) + khh) + HG y) + H H z) + r + k + )[P G y) + P H z) ] + + r + k)n G )n H ) 3

12 Mathematics 07, 5, 6 of 3 Remark From the efinition of HG) an PG), it is obvious that the complete graph has the largest HG) an PG) among all graphs on the same number of vertices So, for any graph G on n vertices we have HG) n ) an PG) n ) + n Also, from the fact that aing an ege to G will increase its aitively weighte Harary inex, it immeiately follows that the complete graph has the largest H A G) among all graph on the same number of vertices Hence, for any graph G on n vertices we have H A G) nn ) From the above remark, we obtain the next corollaries immeiately Corollary 5 Let G be a r-regular graph an H be a k-regular roote graph Then: H A G{H}) r + )[rr + k) + k k + )] + rkr + )[ + r + k Corollary 6 Let G be a r-regular graph an H be a k-regular graph Then: + k 3 ] H A G H)y; z) ) r + )r + k + )k + rk[ + r + k ], ) H A G H)y; z) r + )r + k + )k + rkr + k) 3 + r + k + )[r + )r + ) + k + )k + )] 4 Conclusions In this paper we have investigate the aitively weighte Harary inex for some graph proucts such as splice, link, corona an roote prouct Also we have etermine lower an upper bouns for some of them Author Contributions: This work is a part of Elnaz Ramezani s Master s Thesis which is one uner supervision of Behrooz Khosravi Conflicts of Interest: The authors eclare no conflict of interest References Ranić, M On the characterization of molecular branching J Am Chem Soc 975, 97, Plavšić, D; Nikolić, S; Trinajstić, N; Mihalić, Z On the Harary inex for the characterization of chemical graphs J Math Chem 993,, Ivanciuc, O; Balaban, TS; Balaban, AT Reciprocal istance matrix, relate local vertex invariants an topological inices J Math Chem 993,, Deng, H; Krishnakumari, B; Venkatakrishnan, YB; Balachanran, S Multiplicatively weighte Harary inex of graphs J Comb Optim 06, oi:0007/s Xu, K; Wang, J; Das, KC; Klavar, S Weighte Harary inices of apex trees an k-apex trees Discret Appl Math 05, 89, Xu, K; Das, KC; Trinajstić, N The Harary Inex of a Graph; Springer: Berlin/Heielberg, Germany, 05 7 Xu, K; Das, KC On Harary inex of graphs Discret Appl Math 0, 59, Alizaeh, Y; Iranmanesh, A; Došlić, T Aitively weighte Harary inex of some composite graphs Discret Math 03, 33, Gutman, I; Ruščić, B; Trinajstić, N; Wilcox, CF Graph theory an molecular orbitals XII Acyclic polyenes J Chem Phys 975, 6, Gutman, I; Trinajstić, N Graph theory an molecular orbitals Total π-electron energy of alternant hyrocarbons Chem Phys Lett 97, 7,

13 Mathematics 07, 5, 6 3 of 3 Došlić, T Vertex-Weighte Wiener polynomials for composite graphs Ars Math Contemp 008,, Ashrafi, AR; Došlić, T; Hamzeh, A The Zagreb coinices of graph operations Discret Appl Math 00, 58, c 07 by the authors; licensee MDPI, Basel, Switzerlan This article is an open access article istribute uner the terms an conitions of the Creative Commons Attribution CC BY) license