The edge Szeged index of a molecular graph G is defined as the sum of products mu(e|G)mv(e|G)
over all edges e = uv of G, where mu(e|G) is the number of edges whose distance to vertex u is
smaller than the distance to vertex v, and where mv(e|G) is defined analogously. The aim of this
paper is to compute the edge Szeged index of the Cartesian product of graphs. As a consequence
of our result, the edge Szeged index of Hamming graphs and C4-nanotubes are computed

Ali Reza Ashrafi

Hasan Yousefi-Azari

Ivan Gutman

Mohammad Hosein Khalifeh

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The Edge Szeged Index of Product GraphsMohammad Hosein Khalifeh,a Hasan Yousefi-Azari,a Ali Reza Ashrafi,b and Ivan Gutmanc,*aSchool of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, I. R. IranbDepartment of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, I. R. IrancFaculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, SerbiaRECEIVED JULY 10, 2007; REVISED FEBRUARY 22, 2008; ACCEPTED FEBRUARY 26, 2008The edge Szeged index of a molecular graph G is defined as the sum of products mu(e|G)mv(e|G)over all edges e = uv of G, where mu(e|G) is the number of edges whose distance to vertex u issmaller than the distance to vertex v, and where mv(e|G) is defined analogously. The aim of thispaper is to compute the edge Szeged index of the Cartesian product of graphs. As a conse-quence of our result, the edge Szeged index of Hamming graphs and C4-nanotubes are com-puted.Keywordsedge Szeged indexedge-vertex Szeged indexmolecular graphC4-nanotubeCROATICA CHEMICA ACTACCACAA 81 (2) 277¿281 (2008)ISSN-0011-1643CCA-3241Original Scientific PaperINTRODUCTIONA graph G is defined as a pair G = (V,E), where V =V(G) is a non-empty set of vertices and E = E(G) is a setof edges. Throughout this paper the term »graph« meansfinite graph in which the set of vertices V is finite.Chemical graphs are just graph-based descriptions ofmolecules, with vertices representing the atoms andedges representing the bonds. A numerical invariant as-sociated with a chemical graph, especially if it is ofchemical significance and/or applicability, is called top-ological index.The Wiener index W is the first topological index in-troduced by the chemist Harold Wiener for investigatingboiling points of alkanes.1 After Wiener, many topologi-cal indices were proposed by chemist and also by mathe-maticians. The Szeged index is a topological index re-lated to W, introduced by one of the present authors,2 de-noted here by Szv. To define the Szeged index of a graphG, we assume that e = uv is an edge connecting the verti-ces u and v. Suppose that nu(e|G) is the number of verti-ces of G lying closer to u than to v, and that nv(e|G) is thenumber of vertices of G lying closer to v than to u. Ver-tices equidistant to u and v are not taken into account.Then the Szeged index of the graph G is defined asSzv(G) = Se=uv∈E(G)nu(e|G)nv(e|G). For more informationabout the Szeged index and its mathematical propertiesone should consult the articles.3–7Suppose that u and v are vertices of G. The distanced(u,v) is defined as the length of a minimal path be-tween u and v. If e = xy is an edge of G then d(u,e) =Min{d(u,x),d(u,y)}.In the Ref. 8, an edge version of the Szeged indexwas introduced, named »edge Szeged index«. This newindex is defined as Sze(G) = Semu(e|G)mv(e|G) wheremu(e|G) is the number of edges whose distance to vertexu is smaller than the distance to vertex v, and wheremv(e|G) is defined analogously.* Author to whom correspondence should be addressed. (E-mail: gutman@kg.ac.yu)Throughout this paper our notation is standard andtaken mainly from the books.9–11 Following Imrich andKlav`ar,11 the Cartesian product G ´ H of two graphs Gand H is defined on the Cartesian product V(G) ´ V(H)of the vertex sets of the factors. The edge set E(G ´ H)is the set of all pairs [(u,v),(x,y)] of vertices for which ei-ther u = x and [v,y] ∈ E(H) or [u,x] ∈ E(G) and v = y.Thus V(G ´ H) = V(G) ´ V(H) andE(G ´ H) = {[(u,v),(x,y)] | u = x, [v,y] ∈ E(H), or,[u,x] ∈ E(G), v = y}. Moreover, Kn denotes a completegraph with n vertices and a hypercube of dimension n,Qn, is the Cartesian product of n copies of K2.MAIN RESULTS AND DISCUSSIONFor investigating the mathematical properties of a giventopological index c, it is important to compute c(G ´ H),for every graphs G and H. One should recall that numer-ous molecular graphs are of the form G ´ H. In thepapers12–14 this problem was solved for the Wiener, ver-tex Szeged, and PI index. In this section we continuealong the same lines, and compute the edge Szeged in-dex of product graphs.We first introduce a further »edge-vertex Szeged in-dex« denoted here by Szev. It is defined as Szev(G) =1/2Suv=e∈E(G) [nu(e|G)mv(e|G) + nv(e|G)mu(e|G)], wherenu(e|G), nv(e|G), mu(e|G) and mv(e|G) are defined in Sec-tion 1.Lemma 1. – Suppose G is an acyclic graph with exactlyn vertices. Then Szev(G) = W(G) – n(n – 1)/2.Proof: By a result of Dobrynin and Gutman,15 Sz(G) =W(G) for acyclic graphs G. So,Szev(G) =1/2Suv=e∈E(G)[nu(e|G)mv(e|G) + nv(e|G)mu(e|G)] =1/2Suv=e∈E(G)[nu(e|G)(nv(e|G)–1) + nv(e|G)(nu(e|G)–1)] =1/2Suv=e∈E(G)[2 nu(e|G)nv(e|G) – (nu(e|G)+nv(e|G))] =Szv(G) – n(n – 1)/2 = W(G) – n(n – 1)/2. In order to prove our main result, we assume thatV(G) = {u1,u2,…,ur}, V(H) = {v1,v2,…,vs}, Am ={[(um,vi),(um,vj)] | vivj ∈ E(H)}, 1 ≤ m ≤ r, and Bn ={[(ui,vn),(uj,vn)] | uiuj ∈ E(G)}, 1 ≤ n ≤ s. It is easy to seethat the Am’s and Bn’s are partitions of the edge set ofG ´ H. We also assume that Cm = {(um,vi) | vi ∈V(H)},1 ≤m ≤ r, and Dn = {(ui,vn) | ui ∈V(G)}, 1 ≤ n ≤ s. Clearly,the Cm’s and also Dn’s are two partitions of V(G ´ H).The following Lemma is crucial for our main result.Lemma 2. – With the above-specified notation we have:(i) For every [(up,vk),(up,vl)] ∈ Ap,dG×H((up,vk),[(ut,vi),(ut,vj)]) <dG´H((up,vl),[(ut,vi),(ut,vj)])if and only if dH(vk,[vi,vj]) < dH(vl,[vi,vj]).(ii) dG´H((up,vk), (ui,vt)) < dG´H((up,vl), (ui,vt))if and only if dH(vk,vt) < dH(vl, vt).Proof: (i) and (ii) are immediate consequences of Corol-lary 1.35 in the book.11 Lemma 3. – For every [(up,vk),(up,vl)] ∈ Ap,(i) m(up,vk)([(up,vk),(up,vl)]|G ´ H) =|V(G)|mvk([vk,vl]|H) + |E(G)|nvk([vk,vl]|H),(ii) n(up,vk)([(up,vk),(up,vl)]|G ´ H = |V(G)|nvk([vk,vl]|H).Proof: Directly from the definition, we havemvk([vk,vl]|H = |{[vi,vj] ∈E(H)|dH(vk,[vi,vj]) < dH(vl,[vi,vj])}|whereas by Lemma 2(i),|{[(ut,vi),(ut,vj)] ∈ At|dG ´ H((up,vk),[(ut,vi),(ut,vj)]) <dG ´ H((up,vl),[(ut,vi),(ut,vj)])}| = mvk([vk,vl]|G). Since the Ai’sare disjoint, |{e ∈∪Am|dG ´ H((up,vk),e) < dG ´ H((up,vl),e)}|= |V(G)|mvk([vk,vl]|G). On the other hand, nvk([vk,vl]|H) =|{vt ∈V(H)|dH(vk,vt) < dH(vl,vt)}|. Therefore By Lemma2(ii), |{[(ui,vt),(uj,vt)] ∈ Bt|dG ´ H((up,vk),[(ui,vt),(uj,vt)]) <dG ´ H((up,vl), [(ui,vt),(uj,vt)])}| = degG(ui)nvk([vk,vl]|H).Since the Bi’s are disjoint, |{e ∈∪Bn | dG ´ H((um,vk),e) <dG ´ H((um,vl),e)}| = (1/2Sl≤ i≤ r degG(ui))nvk([vk,vl]|H). But(∪Am) ∩ (∪Bn) = ∅, so m(up,vk)([(up,vk),(up,vl)]|G ´ H) =|{e ∈ (∪Am) ∪ (∪Bn)|d((up,vk),e) < d((up,vl),e)}| =|V(G)| mvk([vk,vl]|H) + |E(G)|nvk([vk,vl]|H).This completes the proof of (i). The proof of (ii) issimilar and so it is omitted. Theorem 1. – Sze(G ´ H) =|V(G)|3Sze(H) + 2|E(G)||V(G)|2Szev(H) +|V(G)||E(G)|2Szv(H) + |V(H)|3Sze(G) +2|E(H)||V(H)|2Szev(G) + |V(H)||E(H)|2Szv(G).Proof: Since (ÈAm) Ç (ÈBn) = Æ,Sze(G ´ H) = Se∈∪Ammu(e|G ´ H) mv(e|G ´ H) +Se∈∪Bnmu(e|G ´ H)mv(e|G ´ H). On the other hand, ByLemma 3Se∈Ammu(e|G ´ H)mv(e|G ´ H) =S[vi,vj]∈E(H)(|V(G)|2mvi([vi,vj]|H)mvj([vi,vj]|H) +|E(G)|2nvi([vi,vj]|H)nvj([vi,vj]|H) +|E(G)||V(G)|(mvj([vi,vj]|H)nvi([vi,vj]|H) +mvi([vi,vj]|H)nvj([vi,vj]|H)).Then we have Se∈∪Ammu(e|G ´ H)mv(e|G ´ H) =|V(G)|3Sze(H) + |E(G)|2|V(G)|Szv(H) +2|V(G)|2|E(G)|Szev(H). Since H ´ G @ G ´ H, for comput-ing Se∈∪Bnmu (e|G ×H)mv(e|G ×H), it suffices to replaceG and H in Eq. (1), which completes our proof. Theorem 2. – Szev(G ´ H) = |V(G)|3Szev(H) +|E(G)||V(G)|2Szv(H) + |V(H)|3Szev(G) +|E(H)||V(H)|2Szv(G).Proof: By definition, we have:278 M. H. KHALIFEH et al.Croat. Chem. Acta 81 (2) 277¿281 (2008)Szev(G ×H) =1/2Se∈∪Am[mu(e|G ×H)nv(e|G ×H) +mv(e|G ×H)nu(e|G ×H] +1/2Se∈∪Bn[mu(e|G ×H)nv(e|G ´ H) +mv(e|G ×H)nu(e|G ´ H)] (2)On the other hand, By Lemma 3,Se∈∪Am[mu(e|G ´ H)nv(e|G ´ H) +mv(e|G ´ H)nu(e|G ´ H)] =Sw∈V(G)Se=uv∈E(H)[2|V(G)||E(G)|nu(e|H)nv(e|H) +|V(G)|2(mu(e|H)nv(e|H) + mv(e|H)nu(e|H)] =|V(G)|3Szev(H) + |E(G)||V(G)|2Szv(H).Similarly, one can compute the second summationof Eq. (2). Suppose ⊗ =in i1 G denotes the Cartesian product ofG1, G2, …, Gn. If G1 = G2 = … = Gn = G then we writeGn as ⊗ =in i1 G . Also, E n ic , = E(⊗ = ≠k k in k1, G ) and Vn ic, =V(⊗ = ≠k k in 1, Gk). Suppose Ei = E(Gi) and Vi = V(Gi), 1 ≤ i ≤n. Then obviously |Vn ic, | = j j in= ≠∏ 1, |Vj | and by Corollary1.35 of the book of Imrich and Klav`ar,11|E(G ´ H)| =12S(a,b)∈V(G´H) degG´H((a,b)) =12Sa∈V(G) Sb∈V(H) [degG(a) + degH(b)] =|E(G)||V(H)| + |E(H)||V(G)|.An inductive argument shows that|E n ic, | = S j j in= ≠1,| | | || |E VVj n,jci⋅(3).Klav`ar, Rajapakse, and Gutman12 proved that Szv(G ´ H) =|V(G)|3Szv(H) + |V(H)|3Szv(G). Again, by means ofmathematical induction one can see that Szv(⊗ =in i1 G ) =S inn icV=1 3| |, Szv(Gi). Therefore, we arrive at the followinggeneralizations of Theorems 1 and 2:Theorem 3. –(i) Szev(⊗ =in i1 G ) =S in=1[|V nc ,1 |3Szev(Gi) + |E n ic , ||V n ic , |2Szv(Gi)],(ii) Sze(⊗ =in iG1 ) =S in=1[|V n ic , |3Sze(Gi) + 2|E n ic , ||V n ic , |2 Szev(Gi)] +|E n ic, |2 |V n ic, |Szv(Gi).Proof: (I) In Theorem 2, we proved the case of n = 2.Suppose the result is valid for n.Szev(⊗ =+in i11 G ) = Szev(Gn+1 ´ ⊗ =in i1 G ) =|Vn+1|3Szev(⊗ =in i1 G ) + |En+1||Vn+1|2Szv(⊗ =in i1 G ) +|V n nc+ +1 1, |3 Szev(Gn+1) + |E n nc+ +1 1, ||V n nc+ +1 1, |2 Szv(Gn+1) =S in=1 |V n ic+1, |3 Szev(Gi) + |V n nc+ +1 1, |3 Szev(Gn+1) +S in=1 |E n ic , ||Vn+1||V n ic+1, |2 Szv(Gi) +S in=1| | | || |E VVn n icn++ +⋅1 11,Szv(Gi) +|E n nc+ +1 1, ||V n nc+ +1 1, |2 Szv(Gn+1) =S in=+11 |V n ic+1, |3 Szev(Gi) +S in=1 |V n ic+1, |2 Szv(Gi) | | | || | | || |V EE VV+1,+1n n ic n n icn+++ ⋅11, +|E n nc+ +1 1, ||V n nc+ +1 1, |2 Szv(Gn+1) . (4)Apply Eq. (3) and our formula for |V n ic, |, we have:|Vn+1||E n ic, | +| | | || |E V +1,n n icnV++⋅11=|Vn+1| S j j in= ≠1,| | | || |E VVj n,jc⋅i+| | | || |E VVn n ,ic+ +⋅1 11n+=S j j in= ≠1,| | | | | || |E V VV+1j n,jci⋅ ⋅n+| | | || |E VVn n ,icn+ ++⋅1 11=S j j in= ≠1,| | | || |E VVj n+ ,jci⋅ 1+| | | || |E VVn n+ ,icn++⋅1 11=S j j in= ≠+11,| | | || |E VVj n+ ,jci⋅ 1.By this equation and Eq. (4), one can see thatSzev(⊗ =+in i11 G ) =S in=+11 [|V n ic+1, |3 Szev(Gi) + |E n ic+1, ||V n ic+1, |2 Szv(Gi)].(II) In Theorem 1, we proved the case of n = 2 andso we can assume that the result is valid for n > 2. ThenSze(Gn+1 ×⊗ in=1Gi) = |Vn+1|3Sze(⊗ in=1Gi) +2|En+1||Vn+1|2Szev(⊗ in=1Gi) +|En+1|2 |Vn+1|Szv(⊗ in=1Gi) + |V n nc+ +1 1, |3Sze(Gn+1) +2|E n nc+ +1 1, ||V n nc+ +1 1, |2 Szev(Gn+1) +|E n nc+ +1 1, |2 |V n nc+ +1 1, | Szv(Gn+1) =S in=1 |V n ic+1, |3 Sze(Gi) + |V n nc+ +1 1, |3 Sze(Gn+1) +2S in=1 |E n ic , ||Vn+1||V n ic+1, |2 Szev(Gi) +2S in=1| | | || |E VVn n+ ,icn++ ⋅1 1 31Szev(Gi) +2 |E n nc+ +1 1, ||V n nc+ +1 1, |2 Szev(Gn+1) +THE EDGE SZEGED INDEX OF PRODUCT GRAPHS 279Croat. Chem. Acta 81 (2) 277¿281 (2008)S in=1 |E n ic , |2 |Vn+1|2 |V n ic+1, |Szv(Gi) +S in=1| | | || |E Vn n+ ,icnv iVSz G++⋅1 2 1 312( ) +2S in=1 |E n ic , ||En+1||V n ic+1, |2 Szv(Gi) +|E n nc+ +1 1, |2 |V n nc+ +1 1, |Szv(Gn+1) =S in=+11 |V n ic+1, |3 Sze(Gi) +2S in=1 |V n ic+1, |2 Szev(Gi) | | | || | | || |V EE VVn n,ic n n icn+++ ++⋅11 11,+2|E n nc+ +1 1, ||V n nc+ +1 1, |2 Szev(Gn+1) +S in=1 |V n ic+1, |Szv(Gi) | | | || | | || |V EE VVn+ n,ic n n+ ,icn+11 11+ ⋅++|E n nc+ +1 1, |2|V n nc+ +1 1, | Szv(Gn+1) =S in=+11[|V n ic+1, |3 Sze(Gi) + 2|E n ic+1, ||V n ic+1, |2 Szev(Gi)] +2 |E n ic+1, |2|V n ic+1, | Szv(Gi).This proves the result. As an immediate consequence of previous theorem,we have:Corollary. –(i) Szev(Gn) = n|V|3n–4(|V|Sev(G) + (n–1) |E|Szv(G)),(ii) Sze(Gn) = n|V|3n–5(|V|2Se(G) + 2(n–1)|V||E|Szev(G) +(n–1)2|E|2Szv(G).We now apply our result to compute the edge Szegedindex of some well-known graphs. Following Imrich andKlav`ar,11 consider the graph G whose vertices are ther-tuples b1b2…br with bi ∈ {0, 1, …, ni–1}, ni ³ 2, andlet two vertices be adjacent if the corresponding tuplesdiffer in precisely one place. Such a graph is called aHamming graph. It is well-known fact that a graph G isa Hamming graph if and only if it can be written in theform G = ⊗ =iN ni1 K . In the following example, the edgeand edge-vertex Szeged indices of a Hamming graph arecomputed.Example 1. – Consider the complete graph Kn. ThenSzv(Kn) = n(n–1)/2, Sze(Kn) = n(n–1)(n–2)2/2 and Szev(Kn) =n(n–1)(n–2)/2.Example 2. – Consider the hypercube Qn. It is easy tosee that Szv(Qn) = n23n–3. Also, by Corollary of Theorem3, Sze(Qn) = n(n–1)223n–5 and Szev(Qn) = n(n–1)23n–4.Example 3. – In this example the edge Szeged index of aC4 nanotube R is computed. By definition of Cartesianproduct of graphs R = Pn ´ Cm, where Pn is a path on nvertices and Cm denotes the cycle on n vertices and m, nare positive integers. To compute the edge Szeged indexof R, we first compute the verted, edge and edge-vertexSzeged indices of Pn and Cm. On can see that, Szv(Pn) =n+13, Sze(Pn) =n+13– (n – 1)2,Szev(Pn) =n+13–n+12,Szv(Cn) =nnn nn32414is evenis odd( )−Sze(Cn) =( )( )( )n nnn nn− −−4 142422is evenis oddandSzev(Cn) =n nnn nn222414( )( )−−is evenis oddTherefore by Theorem 3,Sze(R) =112×16 26 27 1224 23 12 163 3 3 2 3 32 3 3 3 2n m n m n m nn m nm m nmn− + − −+ − − is even23 12 12 3 1226 21 163 2 3 2 23 2 3 3nm n m m nm n mn m n m n− − + + −+ + m n m nm n3 2 3 224 22− − is oddExample 4. – Suppose Hn1,n2,...nr denotes the Hamminggraph with parameters n1, n2, …, nr . By Theorem 3, onecan compute a formula for computing edge Szeged in-dex of Hn1,n2,...nr For the edge-vertex Szeged index of thisgraph, we have Eq. (5)280 M. H. KHALIFEH et al.Croat. Chem. Acta 81 (2) 277¿281 (2008)Szev(Hn1,n2,...nr) = nr rnrnnnnniiri iijiji jri31 2 214441 341 1414= =∏ ∑− + + + + − ,,jririr === ∑∑∑111 . (5)Acknowledgements. – We are indebted to the referees forsuggestions and helpful remarks. One of us (ARA) was par-tially supported by the Center of Excellence of AlgebraicMethods and Applications of the Isfahan University of Tech-nology.REFERENCES1. H. Wiener, J. Am. Chem. Soc. 69 (1947) 17–20.2. I. Gutman,Graph Theory Notes New York 27 (1994) 9-15.3. O. M. Minailiuc, G. Katona, M. V. Diudea, M. Strunje, A.Graovac, and I. Gutman, Croat. Chem. Acta 71 (1998)473-488.4. M. V. Diudea and I. Gutman, Croat. Chem. Acta 71 (1998)21–51.5. P. V. Khadikar, N. V. Deshpande, P. P. Kale, A. Dobrynin, I.Gutman, and G. Dömötör, J. Chem. Inf. Comput. Sci. 35(1995) 547–550.6. A. Dobrynin and I. Gutman, Graph Theory Notes New York28 (1995) 21–23.7. A. A. Dobrynin, Croat. Chem. Acta 70 (1997) 819–825.8. I. Gutman and A. R. Ashrafi, Croat. Chem. Acta, in press.9. N. Trinajsti}, Chemical Graph Theory, 2nd edn, CRCPress, Boca Raton, 1992.10. F. Harary,Graph Theory, Addison-Wesley, Reading, 1994.11. W. Imrich and S. Klav`ar, Product Graphs: Structure andRecognition, Wiley, New York, 2000.12. S. Klav`ar, A. Rajapakse, and I. Gutman, Appl. Math. Lett.9 (1996) 45–49.13. S. Klav`ar, MATCH Commun. Math. Comput. Chem. 57(2007) 573–586.14. M. H. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi,Discr.Appl. Math., in press.15. A. Dobrynin and I. Gutman, Graph Theory Notes New York28 (1995) 21–23.SA@ETAKBridni Segedinski indeks produkta grafovaMohammad Hosein Khalifeh, Hasan Yousefi-Azari, Ali Reza Ashrafi i Ivan GutmanBridni Segedinski indeks molekuskog grafa G je definiran kao zbroj produkata mu(e|G)mv(e|G) preko svihbridova e = uv grafa G, gdje je mu(e|G) broj bridova ~ija je udaljenost od vrha u manja nego udaljenost od vrhav, i gdje je mv(e|G) definiran analogno. U ovom radu odre|en je bridni Segedinski indeks za Kartezijev produktgrafova. Pomo}u ovog rezultata izra~unati su bridni Segedinski indeksi Hammingovih grafova te C4-nanocijevi.THE EDGE SZEGED INDEX OF PRODUCT GRAPHS 281Croat. Chem. Acta 81 (2) 277¿281 (2008)

The Edge Szeged Index of Product Graphs

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