let g be a simple connected graph and {v1, v2, …, vk} be the set of pendent (vertices ofdegree one) vertices of g. the reduced distance matrix of g is a square matrix whose (i,j)–entry is the topological distance between vi and vj of g. in this paper, we obtain the spectrumof the reduced distance matrix of thorn graph of g, a graph which obtained by attaching somenew vertices to pendent vertice...

In this paper we propose and study a new structural invariant for graphs, called distance-unbalancedness, as measure of how much graph is (un)balanced in terms distances. Explicit formulas are presented several classes well-known graphs. Distance-unbalancedness trees also studied. A few conjectures stated some open problems proposed.

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

Let G be a simple, undirected graph. A special network N, called a balanced network, is constructed from G such that maximum matchings and f-factors in G correspond to maximum flows in N. A max-balancedflow-min-balanced-cut theorem is proved for balanced networks. It is shown that Tutte’s Factor Theorem is equivalent to this network flow theorem, and that f-barriers are equivalent to minimum ba...

let $g$ be a connected graph, and let $d[g]$ denote the double graph of $g$. in this paper, we first derive closed-form formulas for different distance based topological indices for $d[g]$ in terms of that of $g$. finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

Polynomial algorithms are given for the following two problems: • given a graph with n vertices and m edges, where m ≥ 3n3/2, find a complete balanced bipartite subgraph with parts about lnn ln(n/m) , • given a graph with n vertices, find a decomposition of its edges into complete balanced bipartite graphs having altogether O(n2/ lnn) vertices. Previous proofs of the existence of such objects, ...

We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with n vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vi...

The concept of balanced decomposition number was introduced by Fujita and Nakamigawa in connection with a simultaneous transfer problem. A balanced colouring of a graph G is a pair (R,B) of disjoint subsets R,B ⊆ V (G) with |R| = |B|. A balanced decomposition D of a balanced colouring C = (R,B) of G is a partition of vertices V (G) = V1 ∪ V2 ∪ . . . ∪ Vr such that G[Vi] is connected and |Vi ∩R|...

We describe a new approach for cluster-based drawing of very large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log...