The lower deg-centric graph of a simple, connected graph \(G\), denoted by \(G_{ld}\), is a graph constructed from \(G\) such that \(V(G_{ld}) = V(G)\) and \(E(G_{ld}) = \{v_iv_j: d_G(v_i,v_j) < \deg_G(v_i)\}\). This paper presents the Roman domination number of lower deg-centric graphs. Also, investigate the properties and structural characteristics of this type of graph.

Let \(K = K(a,p;\lambda_1,\lambda_2)\) be the multigraph with: the number of vertices in each part equal to \(a\); the number of parts equal to \(p\); the number of edges joining any two vertices of the same part equal to \(\lambda_1\); and the number of edges joining any two vertices of different parts equal to \(\lambda_2\). The existence of \(C_4\)-factorizations of \(K\) has been settled when \(a\) is even; when \(a \equiv 1 \ (\mbox{mod } 4)\) with one exception; and for very few cases when \(a \equiv 3 \ (\mbox{mod } 4)\). The existence of \(C_z\)-factorizations of \(K\) has been settled when \(a \equiv 1 \ (\mbox{mod } z)\) and \(\lambda_1\) is even; when \(a \equiv 0 \ (\mbox{mod } z)\); and when \(z=2a\) where both \(a\) and \(\lambda_1\) is even. In this paper, we give a construction for \(C_z\)-factorizations of \(K\) for \(z \in \{ 4,4a \}\) when \(a\) is even.

In 1940, Birkhoff raised the open problem of computing of all posets/lattices on \(n\) elements up to isomorphism for small \(n\). Many authors tried to solve this problem by providing algorithms such as nauty. In 2020, Gebhardt and Tawn given an orderly algorithm for constructing unlabelled lattices of given size and explicitly obtained the number of lattices on up to \(20\) elements. In 2020, Bhavale and Waphare introduced the concept of nullity of a poset as the nullity of its cover graph. Recently, Bhavale and Aware counted lattices having nullity up to two. Bhavale and Aware also counted all non-isomorphic lattices on \(n\) elements, containing up to three reducible elements, having arbitrary nullity \(k \geq 2\). In this paper, we count up to isomorphism the class of all lattices on \(n\) elements containing four comparable reducible elements, and having nullity three.

In this paper, we consider (\(a,b\))- parity factors in graphs and obtain a toughness condition for the existence of (\(a,b\))-parity factors. Furthermore, we show that the result is sharp in some sense.

The most serious physical disability in children is caused by cerebral palsy (CP), a frequent mobility disease in children. Early diagnosis is essential for an early intervention and it helps in potential recovery of infants at high risk. Diagnosis of cerebral palsy is crucial at an early stage since it allows monitoring and therapy sooner. Children with cerebral palsy are prone to high error and may cause underestimation in the values of Hypothalamic–Pituitary–adrenal that is often detected in children with limitations in mobility. Accelerometer-based motion sensors have been acknowledged as the standard for accurately measuring PA in children and adolescents. GAIT aims to map these readings and create a 3-Dimensional model to map coordinates and perform analysis, however finding the severity and the type of Cerebral Palsy is a task due to a lack of classification models. The paper aims to deploy a classification model to predict the presence, intensity and severity of the condition in infants / adults by using the coordinate dataset provided by the GAIT Lab datasets available within the institute’s GAIT Lab. Alongside this, also focusing on deploying an infant cerebral palsy prediction model that can predict the condition in early stages and be used for treatment.

This paper contributes to the classification of non-trivial \(2\)-designs with block size \(5\) admitting a block-transitive automorphism group. Let \({\cal D=(P,B)}\) be a non-trivial 2-\((v,5,\lambda)\) design and \(G\) be a block-transitive automorphism group of \(\mathcal{D}\). The main aim of this paper is to determine all pairs \((\mathcal{D},G)\) when Soc(\(G\)) is a sporadic simple group.

In mathematics education research, mathematics task sets involving mixed practice include tasks from many different topics within the same assignment. In this paper, we use graph decompositions to construct mixed practice task sets for Calculus I, focusing on derivative computation tasks, or tasks of the form “Compute \(f'(x)\) of the function \(f(x)=\) [elementary function].” A decomposition \(D\) of a graph \(G=(V,E)\) is a collection \(\{H_1, H_2, …, H_t\}\) of nonempty subgraphs such that \(H_i=G[E_i]\) for some nonempty subset \(E_i\) of \(E(G)\), and \(\{E_1, E_2, …, E_t\}\) is a partition of \(E(G)\). We extend results on decompositions of the complete directed graph due to Meszka & Skupień to construct balanced task sets that assess the Chain Rule.

Let \(G\) be a graph of order \(n\) and let \(A\) be an additive Abelian group with identity 0. A mapping \(l : V(G) \to A \setminus \{0\}\) is said to be a \(A\)-vertex magic labeling of \(G\) if there exists a \(\mu \in\) \(A\) such that \(w(v) = \sum\limits_{u \in N_G(v)} l(u) = \mu\) for all \(v \in V\) and \(\mu\) is called a magic constant of \(\ell\). The group distance magic set of an \(A\)-vertex magic graph \(gdms(G,A)\) is defined as \(gdms(G,A):= \{ \lambda: \lambda \text{ is a magic constant of some $A$-vertex magic labeling} \}\). In this paper, we investigate under what conditions \(gdms(G,A)\) is a subgroup of \(A\). We also introduce the concept of the reduced group distance magic set, \(rgdms(G, A)\), which can be used as a tool to determine \(gdms(G, A)\).

Let \(2\le k\in\mathbb{Z}\). We say that a total coloring of a \(k\)-regular simple graph via \(k+1\) colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. We prove that Hamming shells of star transposition graphs and Hamming cubes have efficient total colorings. Also in this work, a focus is set upon the graphs of girth \(2k\) and \(k\). Efficient total colorings of finite connected simple cubic graphs of girth 6 are constructed. These are of two specific types, namely: (a) those whose 6-cycles use just 3 colors with antipodal monochromatic pairs of vertices or edges; (b) those whose 6-cycles do not respect item (a) so they use four colors. An orthogonality property holds for all graphs of type (a). Such property allows further edge-half-girth colorings in the corresponding prism graphs.

A \(\{2\}\)-dominating function (\(\{2 \}\)DF) on a graph \(G=(V(G),E(G))\) is a function \(f : V(G) \rightarrow \{0,1,2 \}\) such that \(f(N[v]) \geq 2\) for every \(v \in V(G)\), where \(N[v]\) is the closed neighourhood of \(v\). The \(\{2\}\)-domination number of \(G\) is the minimum weight \(\omega(f) = \sum\limits_{v \in V(G)} f(v)\) among all \(\{2 \}\)-dominating functions on \(G\). In this article, we prove that if \(G\) and \(H\) are graphs with no isolated vertex, then for any vertex \(v \in V(H)\) there are six closed formulas for the \(\{2\}\)-domination number of the rooted product graph \(G \circ_v H\). We also characterize the graph \(G\) and \(H\) that satisfy each of these formulas.