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.

In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.

in this paper, the degree distance and the gutman index of the corona product of two graphs are determined. using the results obtained, the exact degree distance and gutman index of certain classes of graphs are computed.

Given a graph G and a set X ⊆ V (G), the relative Wiener index of X in G is defined as WX(G) = ∑ {u,v}∈(X2 ) dG(u, v) . The graphs G (of even order) in which for every partition V (G) = V1+V2 of the vertex set V (G) such that |V1| = |V2| we have WV1(G) = WV2(G) are called equal opportunity graphs. In this note we prove that a graph G of even order is an equal opportunity graph if and only if it...

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...

A balanced graph is a bipartite graph with no induced circuit of length 2 (mod 4). These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for ...

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

The Mostar index of a graph was defined by Do\v{s}li\'{c}, Martinjak, \v{S}krekovski, Tipuri\'{c} Spu\v{z}evi\'{c} and Zubac in the context study properties chemical graphs. It measures how far given is from being distance-balanced. In this paper, we determine two well-known families graphs: Fibonacci cubes Lucas cubes.