A lower bound on the eccentric connectivity index of a graph

AbstractIn 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, ξC(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

Degree distance and minimum degree

Let $G$ be a finite connected graph of order $n$, minimum degree $\delta$ and diameter $d$. The degree distance $D^\prime(G)$ of $G$ is defined as $\sum_{\{u,v\}\subseteq V(G)}({\rm deg}~ u+{\rm deg}~ v)d(u,v)$, where ${\rm deg}~ w$ is the degree of vertex $w$ and $d(u,v)$ denotes the distance between $u$ and $v$. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that $$D^\prime(G)\le \frac{1}{4}dn\left(n-\frac{d}{3}(\delta+1)\right)^2+O(n^3).$$ As a corollary, we obtain the bound $D^\prime(G)\le \frac{n^4}{9(\delta+1)}+O(n^3)$ for a graph $G$ of order $n$ and minimum degree $\delta$. This result, apart from improving on a result of Dankelmann, Gutman, Mukwembi and Swart [3] for graphs of given order and minimum degree, settles a conjecture of Tomescu [14] completely. Let $G$ be a finite connected graph of order $n$, minimum degree $\delta$ and diameter $d$. The degree distance $D^\prime(G)$ of $G$ is defined as $\sum_{\{u,v\}\subseteq V(G)}(\operatorname{deg} u+\operatorname{deg}v)d(u,v)$, where $\operatorname{deg}w$ is the degree of vertex $w$ and $d(u,v)$ denotes the distance between $u$ and $v$. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that $$D^\prime(G)\le \frac{1}{4}dn\left(n-\frac{d}{3}(\delta+1)\right)^2+O(n^3).$$ As a corollary, we obtain the bound $D^\prime(G)\le \frac{n^4}{9(\delta+1)}+O(n^3)$ for a graph $G$ of order $n$ and minimum degree $\delta$. This result, apart from improving on a result of Dankelmann, Gutman, Mukwembi and Swart [3] for graphs of given order and minimum degree, settles a conjecture of Tomescu [14] completely. doi:10.1017/S000497271200035

Average distance and connected domination

AbstractWe give a tight upper bound on the average distance of a connected graph of given order in terms of its connected domination number. Our results are a strengthening of a result by DeLaViña, Pepper, and Waller [A note on dominating sets and average distance, Discrete Mathematics 309 (2009) 2615–2619] on average distance and connected domination number

Wiener index of trees of given order and diameter at most 6

The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order $n$ and diameter at most $6$. DOI: 10.1017/S000497271300081

Lower bounds on the leaf number in graphs with forbidden subgraphs

Let G be a simple, connected graph. The leaf number, L(G) of G, is dened as the maximum number of leaf vertices contained in a spanning tree of G. Assume that G is a triangle-free graph with minimum degree δ, order n and leaf number L(G). We show that L(G) ≥ δ - 1 /δ + 3n + cδ for δ= 4 and δ= 5, where cδ is a constant that depends on δ only. Similar bounds are shown to hold for triangle-free and C4-free graphs.Mathematics Subject Classication (2010): 05C05.Key words: Leaf number, minimum degree, order, triangle-free graphs

A Treatise of Biological Models

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