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In this paper, we define new graph operations F-composition F (G)[H], where F (G) be one of the symbols S(G),M(G),Q(G),T(G),Λ(G),Λ[G],D2(G),D2[G]. Further, we give some results for the Wiener indices of the these graph operations.

In this paper, we introduce the concept of deficiency sum matrix Sdf (G) of a simple graph G = (V;E) of order n. The deficiency df(v) of a vertex v 2 V is the deviation between the degree of the vertex v and the maximum degree of the graph. The deficiency sum matrix Sdf (G) is a matrix of order n whose (i; j)-th entry isdf(vi) +df(vj), if the vertices vi and vj are adjacent and 0, otherwise. In addition, we introduce deficiency sum energy ESdf (G) of a graph G and establish some bounds for ESdf (G). Further, deficiency sum energy of some classes of graphs are obtained. Moreover, we construct an algorithm and python(3.8) code to find out spectrum and deficiency sum energy of graph G [ABSTRACT FROM AUTHOR]

COVID-19 is a disease caused by a novel coronavirus that is quickly spreading over the world. There is no specific medicine for the treatment of COVID-19, and it is being treated with existing drugs. To treat this condition, new medication research is being conducted. A topological index is a form of molecular descriptor that simply establishes numerical values associated with a compound's molecular structure and is efficiently employed in modeling several physico-chemical aspects in numerous quantitative structure-property/activity relationship (QSPR/QSAR) investigations. In this article, we compute some general temperature topological indices of chloroquine, hydroxychloroquine, theaflavin, lopinavir, ritonavir, and arbidol molecular structures. In addition, QSPR analysis of the relevant topological indices is presented, and it is established that these topological indices are highly correlated with the physico-chemical properties of COVID-19 drugs. This theoretic approach may aid chemists and others working in the pharmaceutical sector in predicting the qualities of COVID-19 drugs before experimenting. [ABSTRACT FROM AUTHOR]

The concept of maximum deficiency matrix Mdf (G) of a simple graph G is introduced in this paper. Let G = (V,E) be a simple graph of order n and let df(vi) be the deficiency of a vertex vi, i = 1, 2, . . ., n, then the maximum deficiency matrix Mdf (G) = [fij ]n×n is defined as: fij = (max{df(vi), df(vj)}, if vivj 2 E(G) 0, otherwise. Further, some coefficients of the characteristic polynomial (G;) of the maximum deficiency matrix of G are obtained. The maximum deficiency energy EMdf (G) of a graph G is also introduced. The bounds for EMdf (G) are established. Moreover, maximum deficiency energy of some standard graphs is shown, and if the maximum deficiency energy of a graph is rational, then it must be an even integer. [ABSTRACT FROM AUTHOR]

In a graph, diametral path is shortest path between two vertices which has length equal to diameter of the graph. Number of diametral paths plays an important role in computer science and civil engineering. In this paper, we introduce the concept of extended transformation graphs. There are 64 extended transformation graphs. We obtain number of diametral paths in some of the extended transformation graphs and we also study the semi-complete property of these extended transformation graphs. Further, a program is given for obtaining number of diametral paths. [ABSTRACT FROM AUTHOR]

A signed graph (or sigraph, in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E), called the underlying graph of S and σ : E →? {+, -} is a function from the edge set E of Su into the set {+, -}, called the signature of S. A sigraph is also denoted as S = (V,E, σ). The splitting sigraph Γ(S) = (VΓ,EΓ, σ Γ) of a sigraph S = (V,E, σ) is formed as follows: For each vertex v ε V(S), take a new vertex v'. Join v' to all vertices of S adjacent to v and define, σ Γ(μv') = σ(μv), μ ε N(v). The sigraph Γ(S) thus obtained is called the splitting sigraph of S. In this paper, we characterize balanced splitting sigraphs and C-consistent splitting sigraphs. Also, we establish a characterization of splitting sigraphs. [ABSTRACT FROM AUTHOR]

A signed graph (or sigraph in short) is an ordered pair S = (Suδ), where Su is a graph G = (V,E), called the underlying graph of S and δ : E → {+,-} is a function from the edge set E of Suinto the set {+,-}, called the signature of S. The x-line sigraph of S denoted by L x (S) is a sigraph defined on the line graph L(Su) of the graph Suby assigning to each edge of L(Su), the product of signs of the adjacent edge ∈ e and f in S. In this paper, first we define semi-total line sigraph and semi-total point sigraph of a given sigraph and then characterize balance and consistency of semi-total line sigraph and semi-total point sigraph. [ABSTRACT FROM AUTHOR]