In this paper, we construct many Hadamard matrices of order \(44\) and we use a new efficient algorithm to investigate the lower bound of inequivalent Hadamard matrices of order \(44\). Using four \((1, -1)\)-circulant matrices of order \(11\) in the Goethals-Seidel array, we obtain many new Hadamard matrices of order \(44\) and we show that there are at least \(6018\) inequivalent Hadamard matrices for this order. Moreover, we use a known method to investigate the existence of double even self-dual codes \([88, 44, d]\) over \(\text{GF}(2)\) constructed from these Hadamard matrices.

Given positive integers \(m, k,\) and \(t\). Let \(D_{m,[k,k+i]} = \{1,2,\ldots,m\} – \{k,k+1,\ldots,k+i\}\). The distance graph \(G(\mathbb{Z}, D_{m,[k,k+i]})\) has vertex set all integers \(\mathbb{Z}\) and edges connecting \(j\) and \(j’\) whenever \(|j-j’| \in D_{m,[k,k+i]}\). The fractional chromatic number, the chromatic number, and the circular chromatic number of \(G(\mathbb{Z}, D_{m,k,i})\) are denoted by \(\chi_f(\mathbb{Z}, D_{m[k,k+i]}), \chi(\mathbb{Z}, D_{m,[k,k+i]}),\) and \(\chi_c(\mathbb{Z}, D_{m,[k,k+i]})\), respectively. For \(i=0\), we simply denote \(D_{m,[k,k+0]}\) by \(D_{m,k}\). \(X(\mathbb{Z}, D_{m,k})\) was studied by Eggleton, Erdős and Skilton [5], Kemnitz and Kolberg [8], and Liu [9], and was completely solved by Chang, Liu and Zhu [1] who also determined \(\chi_c(\mathbb{Z}, D_{m,k})\) for any \(m\) and \(k\). The value of \(\chi_c(\mathbb{Z}, D_{m,k})\) was studied by Chang, Huang and Zhu [2] who finally determined \(\chi_c(\mathbb{Z}, D_{m,k})\) for any \(m\) and \(k\). This paper extends the study of \(G(\mathbb{Z}, D_{m,[k,k+i]})\) to values \(i\) with \(1 \leq i \leq k-1\). We completely determine \(\chi_f(\mathbb{Z}, D_{m,[k,k+i]})\) and \(\chi(\mathbb{Z}, D_{m,k,i})\) for any \(m\) and \(k\) with \(1 \leq i \leq k-1\). However, for \(\chi_c(\mathbb{Z}, D_{m,[k,k+i]})\), only some special cases are determined.

We introduce graphs \(G\) with at least one maximum independent set of vertices \(I\), such that \(\forall v \in V(G) \setminus I\), the number of vertices in \(N_G(v) \cap I\) is constant. When this number of vertices is equal to \(\lambda\), we say that \(I\) has the \(\lambda\)-property and that \(G\) is \(\lambda\)-regular-stable. Furthermore, we extend the study of this property to the well-covered graphs (that is, graphs where all maximal independent sets of vertices have the same cardinality). In this study, we consider well-covered graphs for which all maximal independent sets of vertices have the \(\lambda\)-property, herein called well-covered \(\lambda\)-regular-stable graphs.

Isometric subgraphs of hypercubes are known as partial cubes. Edge-critical partial cubes are introduced as the partial cubes \(G\) for which \(G – e\) is not a partial cube for any edge \(e\) of \(G\). An expansion theorem is proved by means of which one can generate many edge-critical partial cubes. Edge-critical partial cubes are characterized among the Cartesian product graphs. We also show that the \(3\)-cube and the subdivision graph of \(K_4\) are the only edge-critical partial cubes on at most \(10\) vertices.

This paper discusses the enumeration of rooted labelled spanning forests of the complete bipartite graph \(K_{m,n}\).

A set of edges \(D\) in a graph \(G\) is a dominating set of edges if every edge not in \(D\) is adjacent to at least one edge in \(D\). The minimum cardinality of an edge dominating set of \(G\) is the edge domination number of \(G\), denoted by \(D_E(G)\). In this paper, we investigate the edge domination number for the cartesian product of an \(n\)-colorable graph \(G\) and the complete graph \(K_n\).

For graph \(G\) with non-empty edge set, a \((j,k)\)-edge labeling of \(G\) is an integer labeling of the edges such that adjacent edges receive labels that differ by at least \(j\), and edges which are distance two apart receive labels that differ by at least \(k\). The \(\lambda’_{j,k}\)-number of \(G\) is the minimum span over the \((j,k)\)-edge labelings of \(G\). By establishing the equivalence of the edge labelings of \(G\) to particular vertex labelings of \(G\) and the line graph of \(G\), we explore the properties of \(\chi_{j,k}(G)\). In particular, we obtain bounds on \(\lambda’_{j,k}(G)\), and prove that the \(\Delta^2\) conjecture of Griggs and Yeh is true for graph \(H\) if \(H\) is the line graph of some graph \(G\). We investigate the \(\lambda’_{1,1}\)-numbers and \(\lambda_{2,1}\)-numbers of common classes of graphs, including complete graphs, trees, \(n\)-cubes, and joins.

The \(n \times n\) Lah matrix \(L_n\) is defined by \((L_n)_{ij} = l(i, j)\), where \({l}(i, j)\) is the unsigned Lah number. In this paper, we investigate the algebraic properties of \(L_n\), and many important relations between \({L}_n\) and Pascal matrix and Stirling matrix, respectively. In addition, we obtain its exponential expansion and Pascal matrix factorization. Furthermore, we introduce a simple method to find and prove combinatorial identities.

Let \(K_n\) be the complete graph on \(n\) vertices. In this paper, we find the necessary and sufficient conditions for the existence of an \((m_1, m_2, \ldots, m_r)\)-cycle system of \(K_n\), where \(m_i\) (\(1 \leq i \leq r\)) are positive even integers, and \(\sum_{i=1}^{r}m_i = 2^k\) for \(k \geq 2\). In particular, if \(r = 1\) then there exists a cyclic \(2^k\)-cycle system of \(K_n\) if and only if \(2^k\) divides \(|E(K_n)|\) and \(n\) is odd.

In 1948, de Bruijn and Erdős proved that every finite linear space on \(v\) points and with \(6\) lines fulfils the inequality \(b \geq v\), and the equality holds if the linear space is a (possibly degenerate) projective plane. This result led to the problem of classifying finite linear spaces on \(v\) points and with \(b = v + s\) lines, \(s \geq 1\). This paper contains the classification of finite linear spaces on \(v\) points and with \(b = v + 4\) lines.