This paper addresses the enumeration of two types of zero-sum sequences. The first type is defined by constraints on both the number of terms and the values they may assume; the second type is constrained by the number of terms and the total absolute value of the sequence. For both cases, exact enumeration formulas are derived in terms of restricted partition functions, and asymptotic estimates are obtained. In the final section, an algebraic analogue of restricted compositions is introduced, and its enumerative properties are analyzed.

A Fibonacci cordial labeling of a graph \(G\) is an injective function \(f: V(G) \rightarrow \{F_0, F_1, \dots,\\ F_n\}\), where \(F_i\) denotes the \(i^{\text{th}}\) Fibonacci number, such that the induced edge labeling \(f^*: E(G) \rightarrow \{0,1\}\), given by \(f^*(uv) = (f(u) + f(v))\) \((\bmod\ 2)\), satisfies the balance condition \(|e_f(0) – e_f(1)| \le 1\). Here, \(e_f(0)\) and \(e_f(1)\) represent the number of edges labeled 0 and 1, respectively. A graph that admits such a labeling is termed a Fibonacci cordial graph. In this paper, we investigate the existence and construction of Fibonacci cordial labelings for several families of graphs, including Generalized Petersen graphs, open and closed helm graphs, joint sum graphs, and circulant graphs of small order. New results and examples are presented, contributing to the growing body of knowledge on graph labelings inspired by numerical sequences.

An S-packing k-coloring of a graph \(G\) (with \(S=(s_1,s_2,\dots)\) is a non-decreasing sequence of positive integers) is a mapping \(f\) from \(V(G)\) to \(\lbrace 1,\dots , k \rbrace\) (the set of colors) such that for every two distincts vertices \(x\) and \(y\) in \(V(G)\) with \(f(x)=f(y)=i\) the distance between \(x\) and \(y\) in \(G\) is bigger than \(s_i\). The S-packing chromatic number \(\chi_S (G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an S-packing k-coloring. Given a set \(D\subset \mathbb{N}^*\), a distance graph \(G(\mathbb{Z}, D)\) with distance set \(D\) is a graph with vertex set \(\mathbb{Z}\) and two distincts vertices \(u\) and \(v\) are adjacents if \(| u-v | \in D\). In this paper, for \(S=(s,s+1,s+1,\dots)\) with \(s \geq \left\lceil \frac{t}{2} \right\rceil\) we give a lower bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and a lower bound of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) with \(d \geq \left\lceil \frac{t}{2} \right\rceil\), for \(S=(s_1,s_2,\dots, s_i,a,a,\dots)\) with \(a \geq \max( 1 , t-2 )\) we give an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and we determine the exact values of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) and also of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(s\geq \max (\left\lceil \frac{t}{2} \right\rceil, t-3)\) and \(d \geq \max (\left\lceil \frac{t}{2} \right\rceil , t-2)\). And we give a lower and an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(S=(1,s,s,\dots)\) with conditions on \(s\) and \(t\), which in the cases \(s\geq \max (t-2,\left\lceil \frac{t}{2}\right\rceil)\) we determine the exact values of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\).

For a graph \(G=(V,E)\), a pair of vertex disjoint sets \(A_{1}\) and \(A_{2}\) form a connected coalition of \(G\), if \(A_{1}\cup A_{2}\) is a connected dominating set, but neither \(A_{1}\) nor \(A_{2}\) is a connected dominating set. A connected coalition partition of \(G\) is a partition \(\Phi\) of \(V(G)\) such that each set in \(\Phi\) either consists of only a singe vertex with the degree \(\mid V(G)\mid-1\), or forms a connected coalition of \(G\) with another set in \(\Phi\). The connected coalition number of \(G\), denoted by \(CC(G)\), is the largest possible size of a connected coalition partition of \(G\). In this paper, we characterize graphs that satisfy \(CC(G)=2\). Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.

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.