For an arborescence \(A_r\), a directed pathos total digraph \(Q=DPT(A_r)\) has vertex set \(V(Q)=V(A_r)\cup A(A_r)\cup P(A_r)\), = where \(V(A_r)\) is the vertex set, \(A(A_r)\) is the arc set, and \(P(A_r)\) is a directed pathos set of \(A_r\). The arc set \(A(Q)\) consists of the following arcs: \(ab\) such that \(a,b \in A(A_r)\) and the head of \(a\) coincides with the tail of \(b\); \(uv\) such that \(u,v \in V(A_r)\) and \(u\) is adjacent to \(v\); \(au\) \((ua)\) such that \(a\in A(A_r)\) and \(u \in V(A_r)\) and the head (tail) of \(a\) is \(u\); \(Pa\) such that \(a \in A(A_r)\) and \(P \in P(A_r)\) and the arc \(a\) lies on the directed path \(P\); \(P_iP_j\) such that \(P_i, P_j \in P(A_r)\) and it is possible to reach the head of \(P_j\) from the tail of \(P_i\) through a common vertex, but it is possible to reach the head of \(P_i\) from the tail of \(P_j\). For this class of digraphs we discuss the planarity; outerplanarity; maximal outerplanarity; minimally nonouterplanarity; and crossing number one properties of these digraphs. The problem of reconstructing an arborescence from its directed pathos total digraph is also presented.

In this paper, we use Riccati transformation technique to establish some new oscillation criteria for the second order nonlinear dynamic equation with damping on time scales $$(r(t)(x^\Delta(t))^\alpha)^\Delta-p(t)(x^\Delta(t))^\alpha+q(t)f(x(t))=0.$$ Our results not only generalize some existing results, but also can be applied to the oscillation problems that are not covered in literature. Finally, we give some examples to illustrate our main results.

In this work we have described the synthesis of Zinc Oxide nanoparticles through chemical and biological methods. For biological synthesis Aspargillus niger was used. The product obtained was characterized through different analytical techniques like XRD, SEM and EDX. The obtained results were matched with the literature. It was confirmed that the Zinc Oxide nanoparticles can also be prepared from Aspargillus niger.Which may be more ecofriendly and economical compared to other commonly used methods.

For an undirected graph \(G\), a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum \(k\)-flow if the absolute values of edges are less than \(k\). We define the zero-sum flow number of \(G\) as the least integer \(k\) for which \(G\) admitting a zero sum \(k\)-flow. In this paper we gave complete zero-sum flow and zero sum number for octagonal grid, generalized prism and book graph.

Most of molecular structures have symmetrical characteristics. It inspires us to calculate the topological indices by means of group theory. In this paper, we present the formulations for computing the several distance-based topological indices using group theory. We solve some examples as applications of our results.

Topological indices are numerical numbers associated with a graph that helps to predict many properties of underlined graph. In this paper we aim to compute multiplicative topological indices of Circulant graph.

Topological indices are numerical numbers associated with a graph that helps to predict many properties of underlined graph. In this paper we aim to compute multiplicative degree based topological indices of Jahangir graph.

We define fractional transforms \(\mathscr{R}_\mu\) and \(\mathscr{H}_\mu\), \(\mu>0\) on the space \(\mathbb{R}\times\mathbb{R}^n\). First, we study these transforms on regular function spaces and we establish that these operators are topological isomorphisms and we give the inverse operators as integro differential operators. Next, we study the \(L^p\)-boundedness of these operators. Namely, we give necessary and sufficient condition on the parameter \(\mu\) for which the transforms \(\mathscr{R}_\mu\) and \(\mathscr{H}_\mu\) are bounded on the weighted spaces \(L^p([0,+\infty[\times\mathbb{R}^n,r^{2a}dr\otimes dx)\) and we give their norms.

In this paper, we study completete monotonicity properties of certain functions associated with the polygamma functions. Subsequently, we deduce some inequalities involving difference of polygamma functions.