In this paper, we analyze parameter improvement under vertex fusion in a graph G. This is a setting in which a new graph G is obtained after identifying a subset of vertices of G in a single vertex. We are interested in distance parameters, in particular diameter, radius and eccentricity of a vertex v. We show that the corresponding problem is NP-Complete for the three parameters. We also find ...

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ( n 2 )−1∑ {x,y}⊂V (G) dG(x, y), where V (G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that if G is a κ-vertex-connected graph, κ ≥ 3 an odd integer, of order n, then μ(G) ≤ n 2(κ+1) + O(1). Our bound is shown to be...

Given an abelian group A, a graph G = (V, E) is said to have a distance two magic labeling in A if there exists a labeling l : E(G) → A − {0} such that the induced vertex labeling l∗ : V (G) → A defined by l∗(v) = ∑ e∈E(v) l(e) is a constant map, where E(v) = {e ∈ E(G) : d(v, e) < 2}. The set of all h ∈ Z+ for which G has a distance two magic labeling in Zh is called the distance two magic spec...

The edge Szeged index of a connected graph G is defined as the sum of products mu(e|G)mv(e|G) over all edges e = uv of G, where mu(e|G) is the number of edges whose distance to vertex u is smaller than the distance to vertex v, and mv(e|G) is the number of edges whose distance to vertex v is smaller than the distance to vertex u. In this paper, we determine the n-vertex unicyclic graphs with th...

the corona product $gcirc h$ of two graphs $g$ and $h$ isobtained by taking one copy of $g$ and $|v(g)|$ copies of $h$;and by joining each vertex of the $i$-th copy of $h$ to the$i$-th vertex of $g$, where $1 leq i leq |v(g)|$. in thispaper, exact formulas for the eccentric distance sum and the edgerevised szeged indices of the corona product of graphs arepresented. we also study the conditions...

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices the same as original graph. The Distance-Hereditary Vertex Deletion problem asks, given a G on n and an integer k, whether there set S at most k such that $$G-S$$ distance-hereditary. This important due to its connection parameter rank-width because graphs are ...

This paper discusses a distance guarding concept on triangulation graphs, which can be associated with distance domination and distance vertex cover. We show how these subjects are interconnected and provide tight bounds for any n-vertex maximal outerplanar graph: the 2d-guarding number, g2d(n) = ⌊ n 5 ⌋; the 2d-distance domination number, γ2d(n) = ⌊ n 5 ⌋; and the 2d-distance vertex cover numb...