The diameter of an undirected unweighted graph G = (V,E) is the maximum value of the distance from any vertex u to another vertex v for u, v ∈ V where distance i.e. d(u, v) is the length of the shortest path from u to v in G. DAG, is a directed graph without a cycle. We denote the diameter of an unweighted DAG G = (V,E) by δ(G). The stretch of a DAG G is the length of longest path from u to v i...

Let G = (V,A,g) be a strongly connected weighted graph. We say that G is max-balanced if for every cut W, the maximum weight over arcs leaving W equals the maximum weight over arcs entering W. A subgraph H of G is max-sufficient if (or every cut W, the maximum weight over a.rcs of G leaving W is attained at some arc of H. A tower T = (CI,C., ... ,C,) is a sequence of arc-sets of G where C'+ I i...

In $1994,$ degree distance  of a graph was introduced by Dobrynin, Kochetova and Gutman. And Gutman proposed the Gutman index of a graph in $1994.$ In this paper, we introduce the concepts of  multiplicative version of degree distance and the multiplicative version of Gutman index of a graph. We find the sharp upper bound for the  multiplicative version of degree distance and multiplicative ver...

The quadratic embedding constant (QE constant) of a graph is new characteristic value defined through the distance matrix. We derive formulae for QE constants join two regular graphs, double graphs and certain lexicographic product graphs. Examples include complete bipartite wheel friendship completely split graph, some associated to strongly

Let G be a non-bipartite strongly regular graph on n vertices of valency k. We prove that if G has a distance-regular antipodal cover of diameter 4, then k ≤ 2(n + 1)/5 , unless G is the complement of triangular graph T (7), the folded Johnson graph J (8, 4) or the folded halved 8-cube. However, for these three graphs the bound k ≤ (n − 1)/2 holds. This result implies that only one of a complem...

Let D be a connected balanced digraph. The classical distance dijD from vertex i to j is the length of shortest directed path in D. L Laplacian matrix and L†=(lij†) Moore–Penrose inverse L. resistance rijD then defined by rijD≔lii†+ljj†−2lij†. C collection digraphs, each member which finite union form D1∪D2∪....∪Dk where Dt digraph with Dt∩(D1∪D2∪⋯∪Dt−1) being single vertex, for all 1<t≤k. In t...

The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose...