A two-mode network is a type of in which nodes can be divided into two sets such way that links established between different types nodes. The relationship separate entities modeled as bipartite network. In computer networks data transmitted form packets source to destination. Such packet-switched rely on routing protocols select the best path. Configurations these depends acquirements; why one...

The Randić index of a graph G, written R(G), is the sum of 1 √ d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d < D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G) ≥ √ dD d+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and...

The Randić index R(G) of a graph G is defined by R(G) = ∑ uv 1 √ d(u)d(v) , where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n ≥ 3 vertices with the Randić index R(G) and the diameter D(G), R(G) − D(G) ≥ √ 2 − n+1 2 and R(G...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...

Given a connected graph G, the Randić index R(G) is the sum of 1 √ d(u)d(v) over all edges {u, v} of G, where d(u) and d(v) are the degree of vertices u and v respectively. Let q(G) be the largest eigenvalue of the singless Laplacian matrix of G and n = |V (G)|. Hansen and Lucas (2010) made the following conjecture:

ABSTRACTIntroduction: Cancer is caused by genetic abnormalities, such as mutation of ontogenesis or tumor suppressor genes which alter downstream signaling pathways and protein-protein interactions. Comparison of protein interactions in cancerous and normal cells can be of help in mechanisms of disease diagnoses and treatments. Methods: We constructed protein interaction networks of cancerous a...

Connectivity augmentation is a classical problem in combinatorial optimization (see [4, 5]). Given a graph G = (V,E) and a parameter τ ∈ N, add a set of new edges E+ such that the augmented graph G′ = (V,E ∪ E+) is τ -connected (resp., τ -edge-connected). Over planar straightline graphs (PSLGs), it is NP-complete to find the minimum number of edges for τ -connectivity or τ -edge-connectivity au...