Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
RESEARCH
Open Access
Minimum generalized degree distance of n-vertex tricyclic graphs Asma Hamzeh, Ali Iranmanesh* and Samaneh Hossein-Zadeh * Correspondence:
[email protected] Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-137, Tehran, Iran
Abstract In (Hamzeh et al. in Stud. Univ. Babe¸s-Bolyai, Chem., 4:73-85, 2012), we introduced a generalization of a degree distance of graphs as a new topological index. In this paper, we characterize the n-vertex tricyclic graphs which have the minimum generalized degree distance. MSC: 05C12; 05C35; 05C05 Keywords: generalized degree distance; tricyclic graphs; Harary index; join; extremal graphs
1 Introduction Topological indices and graph invariants based on the distances between the vertices of a graph are widely used in theoretical chemistry to establish relations between the structure and the properties of molecules. They provide correlations with physical, chemical and thermodynamic parameters of chemical compounds [–]. The Wiener index is a well-known topological index, which equals to the sum of distances between all pairs of vertices of a molecular graph []. It is used to describe molecular branching and cyclicity and establish correlations with various parameters of chemical compounds. In this paper, we only consider simple and connected graphs. Let G be a connected graph with the vertex and edge sets V (G) and E(G), respectively, and the number of vertices and edges of G are denoted, respectively, by n and m. As usual, the distance between the vertices u and v of G is denoted by dG (u, v) (d(u, v) for short). It is defined as the length of any shortest path connecting u and v in G. We let dG (v) be the degree of a vertex v in G. The eccentricity of v, denoted by ε(v), is the maximum distance from vertex v to any other vertex. The diameter of a graph G is denoted by diam(G) and is the maximum eccentricity over all vertices in a graph G. The radius r = r(G) is defined as the minimum of ε(v) over all vertices v ∈ V (G). A connected graph G with n vertices and m edges is called tricyclic if m = n + . The join G = G + G of two graphs G and G with disjoint vertex sets V and V and edge sets E and E is the graph union G ∪ G together with all the edges joining V and V . Additively weighted Harary index is defined as follows in [] HA (G) =
d– (u, v) dG (u) + dG (v) .
{u,v}⊆V (G)
©2013 Hamzeh et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
Page 2 of 8
There are two papers [, ], which introduced a new graph invariant with the name degree distance. It is defined as follows:
D (G) =
d(u, v) dG (u)+dG (v) .
{u,v}⊆V (G)
Generalized degree distance is denoted by Hλ (G) and defined as follows in []. For every vertex x, Hλ (x) is defined by Hλ (x) = Dλ (x)dG (x), where Dλ (x) = y∈V (G) dλ (x, y), and to avoid confusion, we denote Hλ (x) in graph G with Hλ (x, G). So we have Hλ (G) =
Hλ (x) =
x∈V (G)
Dλ (x)dG (x) =
dλ (u, v) dG (u) + dG (v) ,
{u,v}⊆V (G)
x∈V (G)
where λ is a real number. If λ = , then Hλ (G) = m. When λ = , this new topological index Hλ (G) equals the degree distance index (i.e., the Dobrynin or the Schultz index). The properties of the degree distance index were studied in [–]. Also, if λ = –, then Hλ (G) = HA (G) (see above). The relation of our new index with other intensely studied indices motivated our present (and future) study. Throughout this paper, Kn and K,n– denote the complete and star graphs on n vertices, respectively. Our other notations are standard and taken mainly from [, , ]. Extremal graph theory is a branch of graph theory that studies extremal (maximal or minimal) graphs, which satisfy a certain property. Extremality can be taken with respect to different graph invariants, such as order, size or girth. The problem of determining extremal values and corresponding extremal graphs of some graph invariants is the topic of several papers, for example, see [–, –].
2 Main results In this paper, we characterize all of n-vertex tricyclic graphs which have the minimum generalized degree distance. It is well known that the natural numbers d ≥ d ≥ · · · ≥ dn ≥ are the degrees of the vertices of a tree if and only if ni= di = n – [, ]. The next lemma characterizes connected tricyclic graphs by their degree sequence. Lemma . [] Let n ≥ and G be an n-vertex tricyclic graph. The integers d ≥ d ≥ · · · ≥ dn ≥ are the degrees of the vertices of a graph G if and only if n (i) i= di = n + , (ii) at least four of them are greater than or equal to , (iii) (d , d , . . . , dn ) ∈/ {(n – , , , , , . . . , ), (n – , , , , , . . . , )} for sufficiently large n. Let xi be the number of vertices of degree i of G for ≤ i ≤ n – . If dG (v) = k, then Dλ (v) =
u∈V (G)
dλ (u, v) =
dλ (u, v) +
u∈V (G),d(u,v)=
≥ k + λ (n – k – ) = λ n – k λ – – λ ,
u∈V (G),d(u,v)≥
dλ (u, v)
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
so Hλ (G) =
dG (v)Dλ (v)
v∈V (G)
≥
n–
kxk λ n – k λ – – λ .
k=
We define Fλ (x , x , . . . , xn– ) =
n–
kxk λ n – k λ – – λ .
k=
We obtain the minimum of Fλ (x , x , . . . , xn– ) over all integers numbers x , x , . . . , xn– ≥ , which satisfy one of the conditions of Lemma .. We rewrite Lemma . in terms of the notations above as follows. Lemma . [] Let n ≥ and G be an n-vertex tricyclic graph. The integers x , x , . . . , xn– ≥ are the multiplicities of the degrees of a graph G if and only if n– (i) xi = n, i= n– (ii) i= ixi = n + , (iii) x ≤ n – , (iv) (x , x , . . . , xn– ) ∈/ {(n – , , , , , . . . , , ), (n – , , , , , , . . . , , )} for sufficiently large n. We denote the set of all vectors (x , . . . , xn– ), which satisfy the conditions above, by . Let G be a connected graph with multiplicities of the degrees (x , . . . , xn– ), and let m ≥ , p > , m + p ≤ n – , xm ≥ and xm+p ≥ . Now we consider the transformation of t , which is defined as follows []: t (x , . . . , xn– ) = x , . . . , xn– = (x , . . . , xm– + , xm – , . . . , xm+p – , xm+p+ + , . . . , xn– ). We have xi = xi for i ∈/ {m – , m, m + p, m + p + } and xm– = xm– + , xm = xm – , = xm+p – , xm+p+ = xm+p+ + . Let ≤ m ≤ n – , xm ≥ . Now consider the transformation t defined as follows []:
xm+p
t (x , . . . , xn– ) = x , . . . , xn– = (x , . . . , xm– + , xm – , xm+ + , . . . , xn– ). That is xi = xi for i ∈/ {m – , m, m + } and xm– = xm– + , xm = xm – , xm+ = xm+ + . Lemma . Suppose that λ is a positive integer number, and consider the set of vectors (x , . . . , xn– ). () If (x , . . . , xn– ) ∈ , then t (x , . . . , xn– ) ∈ , unless m = and x = n – , () Fλ (t (x , x , . . . , xn– )) < Fλ (x , x , . . . , xn– ).
Page 3 of 8
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
Page 4 of 8
n– n– n– Proof () We can easily see that n– i= xi = i= xi , i= ixi = i= ixi . If m = , x = n – , then x > n – , and in this case, t (x , . . . , xn– ) ∈/ . Now if x > n – , according to x ≤ n – , we have m = and x = n – . Therefore, we conclude that if (x , . . . , xn– ) ∈ , then x > n – if and only if m = and x = n – . () With a simple calculation, we have Fλ (x , x , . . . , xn– ) – Fλ t (x , x , . . . , xn– ) = λ – (p + ) > . The proof is now completed.
Lemma . Suppose that λ is a positive integer number, and consider the set of vectors (x , . . . , xn– ). () If (x , . . . , xn– ) ∈ , then t (x , . . . , xn– ) ∈ , unless m = and x = n – , () Fλ (t (x , x , . . . , xn– )) < Fλ (x , x , . . . , xn– ). Proof The proof is similar to the proof of the previous lemma, by taking p = .
Theorem . Let n ≥ and λ be a positive integer. If G belongs to the class of connected tricyclic graphs on n vertices, then we have min Hλ (G) = n – n + + λ n + n – , and all the extremal graphs are isomorphic to H or H , where H is obtained by identifying the center of a star K,n– with an arbitrary vertex of a complete graph K , and H is obtained by identifying the center of a star K,n– with an arbitrary four degree vertex of a graph A , which is shown in Figure , respectively. Proof To find minimum Hλ (G), over the classes of connected tricyclic graphs with n vertices, it is enough to find min(x ,x ,...,xn– )∈ Fλ (x , x , . . . , xn– ). At first, we consider the case n = . In this case, only tricyclic graph with vertices is K , that is, Hλ (K ) = , and the theorem is proved in this case. Now let n ≥ . Let us consider n = . The all connected tricyclic graphs with vertices, is A , A , A and A . Meanwhile, Hλ (A ) = + · λ+ = θ (), Hλ (A ) = + · λ+ = θ (), where θ (n) = (n – n + ) + λ (n + n – ), Hλ (A ) = + λ+ , and Hλ (A ) = + λ · , respectively, where Ai (i = , . . . , ) are receipted in Figure . So A and A are extremal graphs, and the theorem is proved in this case.
Figure 1 The graphs A1 , A2 , A3 and A4 .
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
Finally, let n ≥ . If xn– ≥ , consider two different vertices x, y ∈ V (G) such that dG (x) = dG (y) = n – . Since n ≥ , we can choose four different vertices p, q, r, s ∈ V (G) – {x, y}. Actually, the vertices p, q, r, s are all adjacent to x and y, hence G has at least four cycles xypx, xyqx, xyrx, and xysx, which contradicts the hypothesis. Therefore, xn– ≤ . Let us analyze the possible values for x , . . . , xn– . If there exist ≤ i < j ≤ n – such that xi ≥ and xj ≥ , then, by applying t for the positions i and j, we obtain a new vector (x , . . . , xn– ) ∈ , for which Fλ (x , . . . , xn– ) < Fλ (x , . . . , xn– ). Similarly, if there exists ≤ i ≤ n – such that xi ≥ , then by t , we obtain a new degree sequence in , for which Fλ (x , . . . , xn– ) < Fλ (x , . . . , xn– ), that is, xi (i = , . . . , n – ) are not greater than . Let us show that all vectors (x , . . . , xn– ) ∈ , realizing the minimum of Fλ , have x = · · · = xn– = . Actually, suppose that there is an index ≤ i ≤ n – such that xi = and xk = for all ≤ k ≤ n – , k = i. In this case, if x ≥ , we can apply t for positions and i and obtain a smaller value for Fλ . Suppose that x = . Since xn– ∈ {, }, we will separately analyze the two cases: (a) xn– = and (b) xn– = . (a) In this case, xn– = xi = , where i ≥ and x = . We can consider different vertices x, y, p, q, r, s ∈ V (G) such that dG (x) = n – ≥ , dG (y) = i ≥ , p, q, r, s, all are adjacent to x and y, respectively. Meanwhile, x and y are adjacent. We have found four cycles xypx, xyqx, xyrx, and xysx, which contradicts the hypothesis. (b) If xn– = , then x = , xi = ( ≤ i ≤ n – ), and is characterized by the equations x + x = n – , x + x + i = n + , and x ≤ n – . Suppose that x ≥ . In this case, we can use t for position and i and deduce a smaller value for Fλ . Thus x + x = n – and x + x + i = n + . We can obtain x = i – ≤ n – . Moreover, x = n – i + ≥ . In this case, we can use t for position and i and deduce a smaller value for Fλ . To sum up, we have x = · · · = xn– = and xn– ∈ {, }. We will separately analyze the two cases: (a ) xn– = and (b ) xn– = again. (a ) If xn– = , then x + x + x = n + . This equation does not hold. If all x , x , and x are not greater than , x + x + x ≤ , which contradicts the hypothesis n ≥ . If one of x , x , and x is greater than , then by using t for the corresponding position, we can obtain a smaller value for Fλ . (b ) If xn– = , then x + x + x + x = n – and x + x + x + x = n + , namely x + x + x = . If x ≥ , then x + x + x ≥ . If x = , then x = x = and x = n – , which contradicts the condition (iii) in Lemma .. Thus, x = or x = . We now distinguish the following two subcases: Subcase . If x = , then x = – x . All its solutions are x = , x = or x = , x = or x = , x = or x = , x = . In the third and fourth cases, by using t for the position , we can obtain a smaller value for Fλ . In the second case, by using t for the position , we obtain the first case. But the first case cannot be changed, since by using t for the position , the vector (n – , , , , . . . , , ) can be (n – , , , , , . . . , , ), which contradicts Lemma .. Thus, (x , x , . . . , xn– ) = (n – , , , , . . . , , ), and the corresponding unique graph is denoted by H. Subcase . If x = , then x + x = . All its solutions are x = , x = or x = , x = . The first case should be removed. By a similar reasoning as before, the second case cannot be transferred by t . Meanwhile, the first transformation cannot be used. Otherwise, the sequence can be changed into (n – , , , , , , . . . , , ), which contradicts Lemma ..
Page 5 of 8
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
Thus, (x , x , . . . , xn– ) = (n – , , , , , . . . , , ), and the corresponding unique graph is denoted by H . We can easily obtain Hλ (H) and Hλ (H ). Thus, Hλ (H) = Hλ (H ) = (n – n + ) + λ (n + n – ). So, case (b) holds. This completes the proof of Theorem .. In [], the authors proved the following theorem. Theorem . Let n ≥ . If G belongs to the class of connected tricyclic graphs on n vertices, then: (a) If n = , then min D (G) = and the unique extremal graph is isomorphic to K ; (b) If n ≥ , then min D (G) = n + n – , and all the extremal graphs are isomorphic to H or H , where H is obtained by identifying the center of a star K,n– with an arbitrary vertex of a complete graph K , and H is obtained by identifying the center of a star K,n– with an arbitrary degree vertex of a graph A , which is shown in Figure , respectively. If we choose λ = in Theorem ., then we can obtain the same result (Theorem . in []). Theorem . Let G belong to the class of connected graphs on n vertices, and let λ be a positive integer number, then for every n ≥ , we have min Hλ (G) = n – n + n – n + λ , and the unique extremal graph is K,n– . Proof Since for n = the property is trivial, so let n ≥ . If G is a connected graph with n n– vertices, then n– i= xi = n and i= ixi ≥ (n – ). We will find the minimum of Fλ (x , x , . . . , xn– ) over all natural numbers x , x , . . . , n– xn– ≥ satisfying n– i= xi = n and i= ixi ≥ (n–). First, we shall prove that under these conditions, all systems (x , x , . . . , xn– ) of natural numbers reaching min Fλ (x , x , . . . , xn– ) must satisfy (i) and (ii) given below: (i) x + · · · + xn– = n; (ii) x + x + · · · + (n – )xn– = n – . Indeed, suppose that x + x + · · · + (n – )xn– > n – . It follows that there exists a smallest index m, ≤ m ≤ n – such that xm > . We shall define the system (y , . . . , yn– ) such that ym– = xm– + , ym = xm – and yi = xi for every i = m – , m. We have n– i= yi = n, n– n– λ iy = ix – ≥ (n – ) and F (x , x , . . . , x ) – F (y , y , . . . , y ) = (–m + n) + i i λ n– λ n– i= i= (m – ) > . Hence Fλ cannot be minimum for systems (x , x , . . . , xn– ), for which n– i= ixi > (n – ). Since natural numbers d , . . . , dn are the degrees of the vertices of a tree of order n if and only if ni= di = n – , we can consider that the numbers x , x , . . . , xn– are the multiplicities of the degrees , . . . , n – of the vertices of a tree T of order n, and we shall denote also Fλ (T) instead of Fλ (x , x , . . . , xn– ). Hence the domain where Fλ must be minimized is defined by the multiplicities of the degrees , . . . , n – of the vertices of a tree T of order n. If T is not isomorphic to K,n– , i.e., if (x , x , . . . , xn– ) = (n – , , . . . , , ), then diam(T) ≥ . Let z, t ∈ V (T) such that d(z, t) = diam(T). It follows that dT (z) = dT (t) = ,
Page 6 of 8
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
and denote by u and v, u = v, the vertices adjacent with t and z, respectively. Suppose that dT (u) = m + p and dT (v) = m, where m ≥ and p ≥ . Let T be the tree of order n defined in the following way: V (T ) = V (T) and E(T ) = E(T) ∪ {zu} – {zv}. By replacing T with T , the degrees of u and v change: dT (u) = m + p + , dT (v) = m – , and the degrees of n – vertices remain unchanged. We get Fλ (T) – Fλ (T ) = (λ – )(p + ) > , and Fλ (T) cannot be minimum. It follows that Fλ (T) is minimum only if T ∼ = K,n– , i.e., x = n – , x = · · · = xn– = and xn– = , when Fλ (n – , , . . . , , ) = (n – n) + (n – n + )λ . By concluding, min Fλ (x , . . . , xn– ) = (n – n) + (n – n + )λ , and equality holds if and only if x = n – , x = · · · = xn– = and xn– = , i.e., G ∼ = K,n– since Hλ (G) = (n – n) + (n – n + )λ . In [], the authors proved the following theorem. Theorem . Let G belong to the class of connected graphs on n vertices, then for every n ≥ , we have min D (G) = n – n + , and the unique extremal graph is K,n– . If we choose λ = in Theorem ., then we can obtain the same result (Theorem . in []). A Moore graph is a graph of diameter k with girth k + . Those graphs have the minimum number of vertices possible for a regular graph with given diameter and the maximum degree. We first bring the following results in [, ]. Lemma . [] Let G be a connected graph of order n ≥ and size m ≥ , and let λ be a negative integer. Then Hλ (G) ≤ ( – λ )M (G) + λ+ mn – λ+ m, and the equality holds if and only if d ≤ , where d is the diameter of G. Lemma . [] Let G be a triangle- and quadrangle-free graph on n vertices with radius r. Then M (G) ≤ n(n + – r). The equality is valid if and only if G is a Moore graph of diameter or G = C . Theorem . Let G be a triangle- and quadrangle-free graph on n vertices and m edges with radius r, and let λ be a negative integer. Then Hλ (G) ≤ ( – λ )n(n + – r) + λ+ mn – λ+ m. The equality is valid if and only if G is a Moore graph of diameter or G = C . Proof By the lemmas above, we have Hλ (G) ≤ – λ M (G) + λ+ mn – λ+ m ≤ – λ n(n + – r) + λ+ mn – λ+ m. The first equality holds if and only if the diameter of G is at most , and the second one holds if and only if G is a Moore graph of diameter or G = C . So, the equality holds if and only if G is a Moore graph of diameter or G = C . This completes the proof.
Page 7 of 8
Hamzeh et al. Journal of Inequalities and Applications 2013, 2013:548 http://www.journalofinequalitiesandapplications.com/content/2013/1/548
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript. Acknowledgements The authors would like to thank the referee for the valuable comments. Received: 1 May 2013 Accepted: 29 August 2013 Published: 19 Nov 2013 References 1. Diudea, MV, Gutman, I, Jantschi, L: Molecular Topology. Huntington, New York (2001) 2. Trinajsti´c, N: Chemical Graph Theory. CRC Press, Boca Raton (1992) 3. Todeschini, R, Consonni, V: Handbook of Molecular Descriptors. Wiley, New York (2000) 4. Wiener, H: Structural determination of the paraffin boiling points. J. Am. Chem. Soc. 69, 17-20 (1947) 5. Alizadeh, Y, Iranmanesh, A, Došli´c, T: Additively weighted Harary index of some composite graphs. Discrete Math. 313, 26-34 (2013) 6. Dobrynin, AA, Kochetova, AA: Degree distance of a graph: a degree analogue of the Wiener index. J. Chem. Inf. Comput. Sci. 34, 1082-1086 (1994) 7. Gutman, I: Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci. 34, 1087-1089 (1994) 8. Hamzeh, A, Iranmanesh, A, Hossein-Zadeh, S, Diudea, MV: Generalized degree distance of trees, unicyclic and bicyclic graphs. Stud. Univ. Babe¸s-Bolyai, Chem. 4, 73-85 (2012) 9. Dankelmann, P, Gutman, I, Mukwembi, S, Swart, HC: On the degree distance of a graph. Discrete Appl. Math. 157, 2773-2777 (2009) 10. Hossein-Zadeh, S, Hamzeh, A, Ashrafi, AR: Extremal properties of Zagreb coindices and degree distance of graphs. Miskolc Math. Notes 11, 129-137 (2010) 11. Tomescu, I: Unicyclic and bicyclic graphs having minimum degree distance. Discrete Appl. Math. 156, 125-130 (2008) 12. Tomescu, I: Some extremal properties of the degree distance of a graph. Discrete Appl. Math. 98, 159-163 (1999) 13. Harary, F: Graph Theory. Addison-Wesley, Reading (1969) 14. Ashrafi, AR, Došli´c, T, Hamzeh, A: Extremal graphs with respect to the Zagreb coindices. MATCH Commun. Math. Comput. Chem. 65, 85-92 (2011) 15. Bucicovschi, O, Cioabˇa, SM: The minimum degree distance of graphs of given order and size. Discrete Appl. Math. 156, 3518-3521 (2008) 16. Khormali, O, Iranmanesh, A, Gutman, I, Ahmadi, A: Generalized Schultz index and its edge versions. MATCH Commun. Math. Comput. Chem. 64, 783-798 (2010) 17. Moon, JW: Counting Labelled Trees. Canadian Mathematical Monographs, vol. 1. Clowes, London (1970) 18. Senior, JK: Partitions and their representative graphs. Am. J. Math. 73, 663-689 (1951) 19. Zhu, W, Hu, S, Ma, H: A note on tricyclic graphs with minimum degree distance. Discrete Math. Algorithms Appl. 3, 25-32 (2011) 20. Hamzeh, A, Iranmanesh, A, Hossein-Zadeh, S: On the generalization of degree distance of graphs (submitted) 21. Yamaguchi, S: Estimating the Zagreb indices and the spectral radius of triangle-and quadrangle-free connected graphs. Chem. Phys. Lett. 458, 396-398 (2008)
10.1186/1029-242X-2013-548 Cite this article as: Hamzeh et al.: Minimum generalized degree distance of n-vertex tricyclic graphs. Journal of Inequalities and Applications 2013, 2013:548
Page 8 of 8
