For a graph \(G\) and any two vertices \(u\) and \(v\) in \(G\), let \(d_G(u,v)\) denote the distance between them and let \(diam(G)\) be the diameter of \(G\). A multi-level distance labeling (or radio labeling) for \(G\) is a function \(f\) that assigns to each vertex of \(G\) a positive integer such that for any two distinct vertices \(u\) and \(v\), \(d_G(u,v) + |f(u) – f(v)| = diam(G) + 1\). The largest positive integer in the range of \(f\) is called the span of \(f\). The radio number of \(G\), denoted \(rn(G)\), is the minimum span of a multi-level distance labeling for \(G\).

A helm graph \(H_n\) is obtained from the wheel \(W_n\) by attaching a vertex of degree one to each of the \(n\) vertices of the cycle of the wheel. In this paper, the radio number of the helm graph is determined for every \(n \geq 3\): \(rn(H_3) = 13\), \(rn(H_4) = 21\), and \(rn(H_n) = 4n + 2\) for every \(n \geq 5\). Also, a lower bound of \(rn(G)\) related to the length of a maximum Hamiltonian path in the graph of distances of \(G\) is proposed.

In this paper, firstly, we define the generalized \(k\)-Horadam sequence and investigate some of its properties. In addition, by also defining the circulant matrix \(C_n(H)\) whose entries are the generalized \(k\)-Horadam numbers, we compute the spectral norm, eigenvalues, and the determinant of this matrix.

The generating function for \(p\)-regular partitions is given by \(\frac{{(q^p;q^p)}_\infty}{{(q;q)}_\infty}\) .In this paper, we will investigate the reciprocal of this generating function. Several interesting results will be presented, and as a corollary of one of these, we will get a parity result due to Sellers for \(p\)-regular partitions with distinct parts.

Motivated by the results from [J. Li, W. Shiu, W. Chan, The Laplacian spectral radius of some graphs, Linear Algebra Appl. \(431 (2009) 99-103]\), we determine the extremal graphs with the second largest Laplacian spectral radius among all bipartite graphs with vertex connectivity \(k\).

Let \(\omega(K_{1,1,t,}{n})\) be the smallest even integer such that every \(n\)-term graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with \(\sigma(\pi) = d_1+d_2+\cdots+d_n \geq \sigma(K_{1,1,t,}{n})\) has a realization \(G\) containing \(K_{1,1,t,}{n}\) as a subgraph, where \(K_{1,1,t,}{n}\) is the \(1 \times 1 \times t\) complete \(3\)-partite graph. Recently, Lai (Discrete Mathematics and Theoretical Computer Science, \(7(2005), 75-81)\) conjectured that for \(n \geq 2t+4\),

\[\sigma(K_{1,1,t,}{n}) = \begin{cases}
(t+1)(n-1)+2 & \text{if \(n\) is odd or \(t\) is odd,}\\
(t+1)(n-1)+1 & \text{if \(n\) and \(t\) are even.}
\end{cases}\]

In this paper, we prove that the above equality holds for \(n \geq t+4\).

A method called the standard construction generates an algebra from a \(K\)-perfect \(m\)-cycle system. Let \({C}_m^K\) denote the class of algebras generated by \(K\)-perfect \(m\)-cycle systems. For each \(m\) and \(K\), there is a known set \(\Sigma_m^K\) of identities which all the algebras in \({C}_m^K\) satisfy. The question of when \({C}_m^K\) is a variety is answered in [2]. When \({C}_m^K\) is a variety, it is defined by \(\Sigma_m^K\). In general, \({C}_m^K\) is a proper subclass of \({V}(\Sigma_m^K)\), the variety of algebras defined by \(\Sigma_m^K\).

If the standard construction is applied to partial \(K\)-perfect \(m\)-cycle systems, then partial algebras result. Using these partial algebras, we are able to investigate properties of \({V}(\Sigma_m^K)\). We show that the free algebras of \({V}(\Sigma_m^K)\) correspond to \(K\)-perfect \(m\)-cycle systems, so \({C}_m^K\) generates \({V}(\Sigma_m^K)\). We also answer two questions asked in [5] concerning subvarieties of \({V}(\Sigma_m^K)\). Many of these results can be unified in the result that for any subset \(K’\) of \(K\), \({V}(\Sigma_m^{K’})\) is generated by the class of algebras corresponding to finite \(K\)-perfect \(m\)-cycle systems.

We examine designs \( \mathcal{D}_i \) and ternary codes \( C_i \), where \( i \in \{112, 113, 162, 163, 274\} \), constructed from a primitive permutation representation of degree 275 of the sporadic simple group \( M^cL \). We prove that \( \dim(C_{113}) = 22, \quad \dim(C_{162}) = 21, \quad C_{113} \supset C_{162}\) and \( M^cL:2 \) acts irreducibly on \( C_{162} \). Furthermore, we have \( C_{112} = C_{163} = C_{274} = V_{27_5}(GF(3)),\) \(
\text{Aut}(\mathcal{D}_{112}) = \text{Aut}(\mathcal{D}_{163})\) = \(
\text{Aut}(\mathcal{D}_{113}) = \text{Aut}(\mathcal{D}_{162}) =
\text{Aut}(C_{113}) = \text{Aut}(C_{162}) = M^{c}L:2 \) while \( Aut(\mathcal{D}_{274}) = Aut(C_{112}) = Aut(C_{163}) = Aut(C_{274}) = S_{275}. \)
We also determine the weight distributions of \( C_{113} \) and \( C_{162} \) and that of their duals.

The purpose of this paper is to investigate some properties of several \(g\)-Bernstein type polynomials to express the bosonic \(p\)-adic \(q\)-integral of those polynomials on \(\mathbb{Z}_p\).

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that every \(1\)-planar graph without chordal \(5\)-cycles and with maximum degree \(\Delta \geq 9\) is of class one. Meanwhile, we show that there exist class two \(1\)-planar graphs with maximum degree \(\Delta\) for each \(\Delta \leq 7\).

In \([12]\) Quackenbush has expected that there should be subdirectly irreducible Steiner quasigroups (squags), whose proper homomorphic images are entropic (medial). The smallest interesting cardinality for such squags is \(21\). Using the tripling construction given in \([1]\) we construct all possible nonsimple subdirectly irreducible squags of cardinality \(21\) \((SQ(21)s)\). Consequently, we may say that there are \(4\) distinct classes of nonsimple \(SQ(21)s\), based on the number \(n\) of sub-\(SQ(9)s\) for \(n = 0, 1, 3, 7\). The squags of the first three classes for \(n = 0, 1, 3\) are nonsimple subdirectly irreducible having exactly one proper homomorphic image isomorphic to the entropic \(SQ(3)\) (equivalently, having \(3\) disjoined sub-\(SQ(7)s)\). For \(n = 7\), each squag \(SQ(21\)) of this class has \(3\) disjoint sub-\(SQ(7)s\) and \(7\) sub-\(SQ(9)s\), we will see that this squag is isomorphic to the direct product \(SQ(7)\) \(\times\) \(SQ(3)\). For \(n = 0\), each squag \(SQ(21)\) of this class is a nonsimple subdirectly irreducible having three disjoint sub-\(SQ(7)s\) and no sub-\(SQ(9)s\). In section \(5\), we describe an example for each of these classes. Finally, we review all well-known classes of simple \(SQ(21)s\).