Henda C. Swart

On the eccentric connectivity index of a graph

If G is a connected graph with vertex set V, then the eccentric connectivity index of G, @x^C(G), is defined as @?v@?Vdeg(v)ec(v) where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. We obtain an exact lower bound on @x^C(G) in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, precise upper and lower bounds are provided.

Note: The edge-Wiener index of a graph

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index We of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on We in terms of order and size. In particular we prove the asymptotically sharp upper bound We(G)@?2^55^5n^5+O(n^9^/^2) for graphs of order n.

On the degree distance of a graph

If G is a connected graph with vertex set V, then the degree distance of G, D^(G), is defined as @?{u,v}@?V(degu+degv)d(u,v), where degw is the degree of vertex w, and d(u,v) denotes the distance between u and v. We prove the asymptotically sharp upper bound D^(G)@?14nd(n-d)^2+O(n^7^/^2) for graphs of order n and diameter d. As a corollary we obtain the bound D^(G)@?127n^4+O(n^7^/^2) for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].

On equality in an upper bound for the restrained and total domination numbers of a graph

Let G=(V,E) be a graph. A set S@?V is a restrained dominating set (RDS) if every vertex not in S is adjacent to a vertex in S and to a vertex in V@?S. The restrained domination number of G, denoted by @cr(G), is the minimum cardinality of an RDS of G. A set S@?V is a total dominating set (TDS) if every vertex in V is adjacent to a vertex in S. The total domination number of a graph G without isolated vertices, denoted by @ct(G), is the minimum cardinality of a TDS of G. Let @d and @D denote the minimum and maximum degrees, respectively, in G. If G is a graph of order n with @d>=2, then it is shown that @cr(G)==2 achieving this bound that have no 3-cycle as well as those connected graphs with @d>=2 that have neither a 3-cycle nor a 5-cycle. Cockayne et al. [Total domination in graphs, Networks 10 (1980) 211-219] showed that if G is a connected graph of order n>=3 and @D==3 with @D=

Trees with Equal Domination and Restrained Domination Numbers

Let G = (V,E) be a graph and let SV. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γr(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T)=γr(T); (ii) T is a γ-excellent tree and T ≠ K2; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γr(T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.

Diameter and inverse degree

The inverse degree r(G) of a finite graph G=(V,E) is defined as r(G)=@?v@?V1degv. We prove that, if G is connected and of order n, then the diameter of G is less than (3r(G)+2+o(1))lognloglogn. This improves a bound given by Erdos et al. by a factor of approximately 2.

A lower bound on the eccentric connectivity index of a graph

In pharmaceutical drug design, an important tool is the prediction of physicochemical, pharmacological and toxicological properties of compounds directly from their structure. In this regard, the Wiener index, first defined in 1947, has been widely researched, both for its chemical applications and mathematical properties. Many other indices have since been considered, and in 1997, Sharma, Goswami and Madan introduced the eccentric connectivity index, which has been identified to give a high degree of predictability. If G is a connected graph with vertex set V, then the eccentric connectivity index of G, @x^C(G), is defined as @?v@?Vdeg(v)ec(v), where deg(v) is the degree of vertex v and ec(v) is its eccentricity. Several authors have determined extremal graphs, for various classes of graphs, for this index. We show that a known tight lower bound on the eccentric connectivity index for a tree T, in terms of order and diameter, is also valid for a general graph G, of given order and diameter.

Domination with exponential decay

Let G be a graph and [emailxa0protected]?V(G). For each vertex [emailxa0protected]?S and for each [emailxa0protected]?V(G)-S, we define [emailxa0protected]?(u,v)[emailxa0protected]?(v,u) to be the length of a shortest path in if such a path exists, and ~ otherwise. Let [emailxa0protected]?V(G). We define wS(v)[emailxa0protected]?u@?S12^d^@?^(^u^,^v^)^-^1 if [emailxa0protected][emailxa0protected]?S, and wS(v)=2 if [emailxa0protected]?S. If, for each [emailxa0protected]?V(G), we have wS(v)>=1, then S is an exponential dominating set. The smallest cardinality of an exponential dominating set is the exponential domination number, @ce(G). In this paper, we prove: (i) that if G is a connected graph of diameter d, then @ce(G)>=(d+2)/4, and, (ii) that if G is a connected graph of order n, then @ce(G)@?25(n+2).

On the eccentric connectivity index and Wiener index of a graph

Abstract Let G be a finite connected graph of order n and minimum degree δ. The eccentric connectivity index ξc (G) of G is defined as ξc (G) = Σv∊V (G) ecG (v)degG (v), where ecG (x) and degG (x) denote the eccentricity and degree of vertex x in G, respectively. We prove that the eccentric connectivity index of G satisfies , and construct graphs which asymptotically attain the bound. Our bound implies some known results by Došlić, Saheli & Vukičević [4], Morgan, Mukwembi & Swart [11], and Zhou & Du [16]. Further, we also determine upper bounds on the well-studied Wiener index in terms of the eccentric connectivity index.

Eccentric counts, connectivity and chordality

Let ej denote the number of vertices of eccentricity j in a graph. We provide sharp lower bounds on ej in graphs of given connectivity and in chordal graphs of given connectivity.