Closed Neighborhood Degree Sum-Based Topological Descriptors of Graphene Structures

Topological descriptors defined on chemical structures enable understanding the properties and activities of chemical molecules. In this paper, we compute closed neighborhood degree sum-based indices for four different Graphene structures. The cardinality of closed neighborhood degree-based edge partitions for four different Graphene structures is used to compute the closed neighborhood degree sum-based indices.

THE STRUCTURE OF GRAPHS ON n VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2

Let G be an undirected simple graph on n vertices and sigma2(G)=n-2 (degree sum of any two non-adjacent vertices in G is equal to n-2) and alpha(G) be the cardinality of an maximum independent set of G. In G, a vertex of degree (n-1) is called total vertex. We show that, for n>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(n+2)/2, where if alpha(G)=2 then G is a disconnected graph, otherwise G is a connected graph, G contains k total vertices and n-k vertices of degree delta=(n-2)/2, where 0<=k<=(n-2)/2. In particular, when k=0 then G is an (n-2)/2-Regular graph.

Degree sum conditions for hamiltonian index

In this note, we show a sharp lower bound of $$\min \left\{{\sum\nolimits_{i = 1}^k {{d_G}({u_i}):{u_1}{u_2} \ldots {u_k}}} \right.$$ min { ∑ i = 1 k d G ( u i ) : u 1 u 2 … u k is a path of (2-)connected G on its order such that (k-1)-iterated line graphs Lk−1(G) are hamiltonian.

Degree based and neighbourhood degree-sum based topological indices of PAH(Dimer 1) in graphene context

In this paper, twenty degree-based topological indices and seven neighbourhood degree-sum-based topological indices of Dimer 1 (two units of chrysene) [4] 0D & 1D in the graphene context are enumerated. The Oligomer Approach[3] is practiced here to explore the interconnection between PAH ( cove type periphery based on 11, 11’-dibromo-5,5’-bis chrysene as a key monomer-Dimer 1) and graphene numerically through the indices.

Degree Sum Exponent Distance Energy of Some Graphs

The degree sum exponent distance matrix M(G)of a graph G is a square matrix whose (i,j)-th entry is (di+dj)^ d(ij) whenever i not equal to j, otherwise it is zero, where di is the degree of i-th vertex of G and d(ij)=d(vi,vj) is distance between vi and vj. In this paper, we define degree sum exponent distance energy E(G) as sum of absolute eigenvalues of M(G). Also, we obtain some bounds on the degree sum exponent distance energy of some graphs and deduce direct expressions for some graphs.