A complex unit gain graph ( T -gain graph) Φ = G , φ ) is a where the function assigns number to each orientation of an edge and its inverse assigned opposite orientation. balanced if product gains oriented cycle (if any) 1. We propose two notions distance matrices D < max min Φ, for any ordering ‘<’ vertex set. characterize graphs which are independent ordering. show holds standard vertices on...

The packing of different circles in a circular container under balancing and distance conditions is considered. Two problems are studied: the first minimizes container’s radius, while second maximizes minimal between circles, as well boundary container. Mathematical models solution strategies provided illustrated with computational results.

Given a connected graph $$G = (V, E)$$ , we seek to partition the vertex set V into k non-empty parts such that subgraph induced by each part is connected, and maximally balanced in way maximum cardinality of these minimized. We refer this problem as min-max denote it -BGP. The vertex-weighted version on trees has been studied since about four decades ago, which admits linear time exact algorit...

A graph G = (V,E) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : V (G) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ V (G), xy ∈ E(G), and the total number of 0, 1 and 2 are balanced. That is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). In thi...