In this paper we obtain coefficient characterization, extreme points and Distortion bounds for the classes of harmonic p-valent functions defined by certain modified operator. Some of our results improve and generalize previously known results.

Financial Mathematics is the application of mathematical methods to financeial problems. It is shown that Harmonic Univalent Functions play important roles in Financial Mathematics. This paper introduced and established a subclass of harmonic univalent functions involving the argument of complex-value functions of the form f = h + 𝑔 . Additionally, this study investigates some properties of this subclass such as necessary and sufficient coefficient bounds, extreme points, Distortion bounds and Hadamard product.

The Sombor index of the graph G is a degree based topological index, deffined as SO ∑,uvϵ,E(G) ,d2u + d2v, where du is the degree of the vertex u, and E(G) is the edge set of G. bounds on SO are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number.

An (n, r)-arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. The maximum size of an (n, r)-arc in PG(2, q) is denoted by mr(2, q). In this paper thirteen new (n, r)-arc in PG(2, 29) and a table with the best known lower and upper bounds on mr(2, 29) are presented. The results are obtained by non-exhaustive local computer search.

Let G (V,σ, ,μ, ) be a fuzzy graph of order n, where σ,(u) is the vertex membership, μ,(u,v) is membership value of an edge and μ,(u) is the strength of vertex. The _rst fuzzy Zagreb index is the sum σ,(ui) μ,(ui) + σ,(uj ) μ,(uj ) where uiuj ,μ,and the corresponding fuzzy Zagreb matrix is the square matrix of order n whose (I, j)th entry whenever i ≠,j, is σ,(ui) μ,(ui)+ σ,(uj ) μ,(uj ) and zero otherwise. In this paper, we introduce the Zagreb Estrada index of fuzzy graphs and establish some bounds for it.

A subset $D$ of the vertex set $V(G)$ in a graph $G$ is a point-set dominating set (or, in short, psd-set) of $G$ if for every set $S\subseteq V- D$, there exists a vertex $v\in D$ such that the induced subgraph $\langle S\cup \{v\}\rangle$ is connected.  The minimum cardinality of a psd-set of $G$ is called the point-set domination number of $G$. In this paper, we establish two sharp lower bounds for point-set domination number of a graph in terms of its diameter and girth. We characterize graphs for which lower bound of point set domination number is attained in terms of its diameter. We also establish an upper bound and give some classes of graphs which attains the upper bound of point set domination number.

The graph energy is the sum of absolute values of the eigenvalues of the (0; 1)-adjacency matrix. Oboudi recently obtained lower bounds for graph energy, depending on the largest and smallest graph eigenvalue. In this paper, a few more Oboudi-type bounds are deduced.

Let $G=(V, E)$ be a simple graph. A set $C$ of vertices $G$ is an identifying set of $G$ if for every two vertices $x$ and $y$ belong to $V$ the sets $N_{G}[x] \cap C$ and $N_{G}[y] \cap C$ are non-empty and different. Given a graph $G,$ the smallest size of an identifying set of $G$ is called the identifying code number of $G$ and is denoted by $\gamma^{ID}(G).$ Two vertices $x$ and $y$ are twins when $N_{G}[x]=N_{G}[y].$ Graphs with at least two twin vertices are not identifiable graphs. In this paper,  we present three bounds for identifying code number.