A graph G of order n is called t-edge-balanced if G satisfies the property that there exists a positive λ for which every graph of order n and size t is contained in exactly λ distinct subgraphs of Kn isomorphic to G. We call λ the index of G. In this article, we obtain new infinite families of 2-edge-balanced graphs. C ⃝ 2013 Wiley Periodicals, Inc. J. Combin. Designs 22: 291–305, 2014

Let G be a balanced bipartite graph of order 2n and minimum degree (G) 3. If for every balanced independent set S of four vertices in G the neighborhood union jN(S)j > n then G is traceable, the circumference is at least 2n ? 2 and G contains a 2-factor (with one exceptional graph for the latter statement). If jN(S)j > n + 2 then G is hamiltonian.

the $k$-th semi total point graph $r^k(g)$ of a graph $g$, is a graph obtained from $g$ by adding $k$ vertices corresponding to each edge and connecting them to endpoint of edge considered. in this paper, we obtain formulae for the resistance distance and kirchhoff index of $r^k(g)$.

We consider graphs without loops or parallel edges in which every edge is assigned + or −. Such a signed graph is balanced if its vertex set can be partitioned into parts V1 and V2 such that all edges between vertices in the same part have sign + and all edges between vertices of different parts have sign − (one of the parts may be empty). It is well-known that every connected signed graph with...

A connected graph $\G$ is called {\em nicely distance--balanced}, whenever there exists a positive integer $\gamma=\gamma(\G)$, such that for any two adjacent vertices $u,v$ of are exactly $\gamma$ which closer to $u$ than $v$, and $v$ $u$. Let $d$ denote the diameter $\G$. It known $d \le \gamma$, distance-balanced graphs with $\gamma = d$ precisely complete cycles length $2d$ or $2d+1$. In th...

the average distance of a vertex $v$ of a connected graph $g$is the arithmetic mean of the distances from $v$ to allother vertices of $g$. the proximity $pi(g)$ and the remoteness $rho(g)$of $g$ are defined as the minimum and maximum averagedistance of the vertices of $g$. in this paper we investigate the difference between proximity or remoteness and the classical distanceparameters diameter a...

The average distance of a vertex $v$ of a connected graph $G$is the arithmetic mean of the distances from $v$ to allother vertices of $G$. The proximity $pi(G)$ and the remoteness $rho(G)$of $G$ are defined as the minimum and maximum averagedistance of the vertices of $G$. In this paper we investigate the difference between proximity or remoteness and the classical distanceparameters diameter a...