On the Gutman index and minimum degree Jaya Percival Mazorodze 1 Department of Mathematics University of Zimbabwe, Harare, Zimbabwe mazorodzejaya@gmail.com Simon Mukwembi School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban, South Africa Mukwembi@ukzn.ac.za Tomáš Vetrík Department of Mathematics and Applied Mathematics University of the Free State, Bloemfontein, South Africa tomas.vetrik@gmail.com Abstract The Gutman index GutG) of a graph G is defined as, {x,y} V G) where V G) is the vertex set of G, degx), degy) are the degrees of vertices x and y in G, and dx, y) is the distance between vertices x and y in G. We show that for finite connected graphs of order n and minimum degree δ, where δ is a constant, GutG) 4 5 5 δ+1) n5 + On 4 ). Our bound is asymptotically sharp for every δ and it extends results of Dankelmann, Gutman, Mukwembi and Swart The edge-wiener index of a graph, Discrete Math. 09 009) 45 457 and Mukwembi On the upper bound of Gutman index of graphs, MATCH Commun. Math. Comput. Chem. 68 01) 4 48, whose bound is sharp only for graphs of minimum degree. Keywords: Gutman index, minimum degree 1 Introduction Graph indices have been studied for decades because of their extensive applications in chemistry. With Wiener s discovery of a close correlation between the boiling points of certain alkanes and the sum of the distances between vertices in graphs representing their molecular structures, it became apparent that graph indices can potentially be used to predict properties of chemical compounds. Like Wiener s original index, Gutman index is also based on distances between vertices of graphs. The Gutman index is a natural extension of the Wiener index. The Gutman index of a finite connected graph G is defined as GutG) =, {x,y} V G) 1 This work will be part of the first author s PhD Thesis. 1
where V G) is the vertex set of G, degx), degy) are the degrees of vertices x and y in G, and dx, y) is the distance between vertices x and y in G. The Gutman index has been studied for example in 1,, 4 and 5. In 5 it was presented that for acyclic structures, the Gutman index reflects exactly the same structural features as the Wiener index. The question, whether theoretical investigations on the Gutman index focusing on the more difficult polycyclic molecules can be done, was posed. Feng studied the Gutman index for unicyclic graphs, and Feng and Liu 4 considered bicyclic graphs in their research. Dankelmann, Gutman, Mukwembi and Swart showed that if G is a connected graph of order n, then GutG) 4 n 5 5 5 + On 9 ). Mukwembi 6 improved this upper bound and presented the result GutG) 4 n 5 5 5 + On 4 ), which shows that On 9 ) can be replaced by On 4 ). However all graphs which attain this bound are of minimum degree, which was motivation for us to study the Gutman index of graphs of minimum degree δ, where δ. In this note we improve the bound of Mukwembi for δ, and show that GutG) 4 5 5 δ + 1) n5 + On 4 ), where n is the order of the graph G and the minimum degree δ is a constant. Moreover we show that our bound is asymptotically sharp for every δ. Results First we present Lemma.1, which will be used in the proof of our main result. Lemma.1 Let G be a connected graph of order n, diameter d and minimum degree δ, where δ is a constant. Let v, v be any vertices of G. 1) Then degv) n 1 dδ + 1) + δ. ) If dv, v ), then degv) + degv ) n 1 dδ + 1) + 4δ. Proof. Let P : v 0, v 1,..., v d be a diametric path of G. Let S V P ) be the set { d 1 } S := v i+1 : i = 0, 1,,...,. For each u S, choose any δ neighbours u 1, u,..., u δ of u and denote the set {u, u 1, u,..., u δ } by P u. Let P = u S P u. Then ) P = δ + 1) + 1.
Let v be any vertex of G. We denote by Nv the closed neighbourhood of v, which is the set that consists of v and its neighbours. Note that if v / P, then v can be adjacent to at most one vertex in S and to neighbors of at most vertices of S, hence v is adjacent to at most δ + 1 vertices in P. If v P, then it can be checked that v can be adjacent to at most δ vertices in P. In both cases we obtain P Nv δ + 1 which implies n P + Nv P Nv ) δ + 1) + 1 + degv) + 1) δ + 1) δ + 1) d + degv) δ. Rearranging the terms, we obtain degv) n 1 dδ + 1) + δ, which completes the proof of 1). Now we prove the statement ). If v, v are any two vertices of G, such that dv, v ), then Nv Nv =. It follows that n P + Nv + Nv P Nv P Nv ) δ + 1) + 1 + degv) + 1) + degv ) + 1) δ + 1) δ + 1) d + degv) + degv ) 4δ, which implies degv) + degv ) n 1 dδ + 1) + 4δ. Now we present our main result. Theorem.1 Let G be a connected graph of order n and minimum degree δ, where δ is a constant. Then and this bound is asymptotically sharp. GutG) 4 5 5 δ + 1) n5 + On 4 ), Proof. We denote the diameter of G the largest distance between any two vertices in G) by d. Let P : v 0, v 1,..., v d be a diametric path of G and let S V P ) be the set S := { v i+1 : i = 0, 1,,..., d 1 For each v S, choose any δ neighbours u 1, u,..., u δ of v and denote the set {v, u 1, u,..., u δ } by P v. Let P = v S P v. Then }. ) P = δ + 1) + 1. 1)
Now let V = {{x, y} : x, y V }. We partition V as follows: where V = P A B, P := {{x, y} : x P and y V G)}, A := {{x, y} V P : dx, y) } and B := {{x, y} V P : dx, y) }. Setting A = a, B = b, we have n ) = P + a + b, and so from 1), a + b = ) n P = 1 ) ) n δ + 1) + 1 n δ + 1) + 1 1. ) Note that GutG) = + +. We bound each term separately. Claim.1 Assume the notation above. Then On 4 ). Proof of Claim.1: We partition S as S = S 1 S, where S 1 = {v j S : j 1 mod 6)}, and S = S S 1. It follows that P = v S1 P v) v S P v). For each vertex x in P, define the score sx) of x as sx) := y V. Then x P sx) = x v S1 P v) sx) + x v S P v) sx). We now consider v S1 P v. For each u, v S 1, u v, we have P u P v = and the neighbourhoods of P u and P v are also disjoint. Write the elements of S 1 as S 1 = {w 1, w,..., w S1 }. For each w j S 1, let P w j = {w j, w j 1, wj,..., wj δ }, where wj 1, wj,..., wj δ are neighbours of w j. Then n degw 1 ) + 1) + degw ) + 1) + + degw S1 ) + 1) 4
and for t = 1,,..., δ, Summing we get That is, Similarly, n degw 1 t ) + 1) + degw t ) + 1) + + degw S1 t ) + 1). δ + 1)n x u S1 P u) x u S P u) x u S1 P u) degx) + δ + 1) S 1. Now from Lemma.1, for every x P, we have ) sx) = degx) degy)dx, y) degx) degx) δ + 1)n δ + 1) S 1. ) degx) δ + 1)n δ + 1) S. 4) y V y V ) n 1 ) dδ + 1) + δ d degx) dn n 1 ) ) dδ + 1) + δ. This, in conjunction with ), 4) and the fact that δ is a constant, yields degx)degy)d G x, y) + x v S1 P v) x v S P v) degx) dn n 1 ) dδ + 1) + δ degx) dn n 1 ) dδ + 1) + δ = dn n 1 ) dδ + 1) + δ x v S1 P v) degx) + x v S P v) degx) dn n 1 ) dδ + 1) + δ δ + 1)n δ + 1) S 1 + δ + 1)n δ + 1) S = dn n 1 ) d 1 dδ + 1) + δ δ + 1)n δ + 1) = On 4 ), as required and so Claim.1 is proven. Now we bound those pairs of vertices, which are in B. ) 5
Claim. Assume the notation above. Then On 4 ). Proof of Claim.: Note that if {x, y} B, then dx, y). This, together with Lemma.1 and the fact that b = On ), gives as claimed. n 1 ) dδ + 1) + δ = b n 1 ) dδ + 1) + δ = On 4 ), Finally, we study pairs of vertices, which are in A. Claim. Assume the notation above. Then 1 16 d n 1 dδ + 1) ) 4 + On 4 ). Proof of Claim.: Let {w, z} be a pair in A, such that degw) + degz) is maximum. Let degw)+degz) = s. Since degw)degz) 1 4 degw)+degz)), we have degw)degz) 1 4 s. 5) Now we find an upper bound on a, the cardinality of A. From ) we have a = 1 n δ + 1) ) + 1 n δ + 1) ) + 1 1 b. 6) Note that all pairs {x, y}, x, y Nw P and all pairs {x, y}, x, y Nz P where Nw Nz) is the closed neighbourhood of w of z)) are in B. Since w and z) can be adjacent to at most one vertex in S and to neighbours of at most vertices of S, it follows that w and z) is adjacent to at most δ + 1 vertices in P. Then we have ) ) degw) δ degz) δ b + = 1 degw) + degx) ) 4δ + 1 degw) + degz)) + 4δ + δ) 1 4 s 4δ + 1 s + 4δ + δ). 6
Hence from 6), we get a 1 n δ + 1) 1 4 s + 4δ + 1 s 4δ + δ). From 5), we have ) + 1 n δ + 1) ) + 1 1 s d 4 s d 1 ) ) n δ + 1) + 1 n δ + 1) + 1 1 4 1 4 s + 4δ + 1 s 4δ + δ). By Lemma.1, s n 1 dδ + 1) + 4δ. Subject to this condition s d 1 d 1 d 1 4 n δ+1) +1) n δ+1) +1) 1 1 4 s + 4δ+1 s 4δ +δ) is maximized for s = n 1 dδ + 1) + O1) to give = d n 1 ) 1 4 dδ + 1) d 16 n 1 dδ + 1) ) 4 + On 4 ). The proof of Claim. is complete. n 1 dδ + 1) ) 1 4 n 1 dδ + 1) ) + On) Now we can complete the proof of the theorem. From Claims.1,. and., we get GutG) = + + The term 1 n 16 d 1 ) 4 dδ + 1) + On 4 ) + On 4 ) + On 4 ) = 1 16 d n 1 ) 4 dδ + 1) + On 4 ). 1 16 d n 1 ) 4 dδ + 1) 7
is maximized, with respect to d, for d = n 5δ+1) to give GutG) 4 5 5 δ + 1) n5 + On 4 ), as desired. It remains to show that the bound is asymptotically sharp. We construct the graph G n,d,δ for d 1 mod ). Let V G) = V 0 V 1... V d, where δ 1 if i mod ), 1 V i = n 1 d 1)δ + 1)) if i = 0, 1 n 1 d 1)δ + 1)) if i = d, 1 otherwise. Let two distinct vertices v V i, v V j be adjacent if and only if j i 1, and let d = n 5δ+1) be an integer. Then the graph G n n, 5δ+1),δ has order n, minimum degree δ and the Gutman index is 4 5 5 δ+1) n5 + On 4 ). Note that if δ =, we obtain the result of 6 as a corollary of our Theorem.1. Corollary.1 Let G be a connected graph of order n. Then References GutG) 4 5 5 n5 + On 4 ). 1 V. Andova, D. Dimitrov, J. Fink and R. Škrekovski, Bounds on Gutman index, MATCH Commun. Math. Comput. Chem. 67 01) 515 54. P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, The Edge- Wiener index of a graph, Discrete Math. 09 009) 45 457. L. Feng, The Gutman index of unicyclic graphs, Discrete Math. Algorithms Appl. 4 01) 669 708. 4 L. Feng, W. Liu, The maximal Gutman index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 66 011) 8 pages. 5 I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput.Sci. 4 1994) 1087 1089. 6 S. Mukwembi, On the upper bound of Gutman index of graphs, MATCH Commun. Math. Comput. Chem. 68 01) 4 48. 8