Strongly and Nicely Edge Distance-Balanced Graphs

Strongly and Nicely Edge Distance-Balanced Graphs

A nonempty graph G is called nicely edge distance-balanced (NEDB), whenever there exists a positive integer γ′G, such that for any edge say e = ab we have: mGa(e) = mGb(e) = γ′G. Which mGa(e) denotes the number of edges laying closer to the vertex a than vertex b and mGb(e) is defined analogously. Also, a nonempty graph G is strongly edge distance-balanced, for every edge say e = ab of G and every i ≥ 0 the number of edges at distance i from a and at distance i + 1 from b is equal to the number of edges at distance i + 1 from a and at distance i from b. In this paper, first we study on some properties of strongly edge distance-balanced graphs. Later, we discuss on some operations of graphs and at last by the help of definition of SEDB graph, classify the NEDB graphs with γ’G = 3.

A nonempty graph G is called nicely edge distance-balanced (NEDB), whenever there exists a positive integer γ′G, such that for any edge say e = ab we have: mGa(e) = mGb(e) = γ′G. Which mGa(e) denotes the number of edges laying closer to the vertex a than vertex b and mGb(e) is defined analogously. Also, a nonempty graph G is strongly edge distance-balanced, for every edge say e = ab of G and every i ≥ 0 the number of edges at distance i from a and at distance i + 1 from b is equal to the number of edges at distance i + 1 from a and at distance i from b. In this paper, first we study on some properties of strongly edge distance-balanced graphs. Later, we discuss on some operations of graphs and at last by the help of definition of SEDB graph, classify the NEDB graphs with γ’G = 3.