A \((g,k; \lambda)\)-difference matrix over the group \((G, o)\) of order \(g\) is a \(k\) by \(g\lambda\) matrix \(D = (d_{ij})\) with entries from \(G\) such that for each \(1 \leq i < j \leq k\), the multiset \(\{d_{il}\) o \(d_{jl}^{-1} \mid 1 \leq l \leq g\lambda\}\) contains every element of \(G\) exactly \(\lambda\) times. Some known results on the non-existence of generalized Hadamard matrices, i.e., \((g,g\lambda; \lambda)\)-difference matrices, are extended to \((g, g-1; \lambda)\)-difference matrices.

The notion of convexity in graphs is based on the one in topology: a set of vertices \(S\) is convex if an interval is entirely contained in \(S\) when its endpoints belong to \(S\). The order of the largest proper convex subset of a graph \(G\) is called the convexity number of the graph and is denoted \(con(G)\). A graph containing a convex subset of one order need not contain convex subsets of all smaller orders. If \(G\) has convex subsets of order \(m\) for all \(1 \leq m \leq con(G)\), then \(G\) is called polyconvex. In response to a question of Chartrand and Zhang [3], we show that, given any pair of integers \(n\) and \(k\) with \(2 \leq k < n\), there is a connected triangle-free polyconvex graph \(G\) of order \(n\) with convexity number \(k\).

In this work, \(\Gamma\) denotes a finite, simple, and connected graph. The \(k\)-excess \(e_k(H)\) of a set \(H \subseteq V(\Gamma)\) is defined as the cardinality of the set of vertices that are at distance greater than \(k\) from \(H\), and the \(k\)-excess \(e_k(h)\) of all \(A\)-subsets of vertices is defined as

\[e_k(h) = \max_{H \subset V(\Gamma),|H|=h} \{ e_k(H) \}\]

The \(k\)-excess \(e_k\) of the graph is obtained from \(e_k(h)\) when \(h = 1\). Here we obtain upper bounds for \(e_k(h)\) and \(e_k\) in terms of the Laplacian eigenvalues of \(\Gamma\).

Let \(G\) be a \(k\)-connected graph and let \(F\) be the simple graph obtained from \(G\) by removing the edge \(xy\) and identifying \(x\) and \(y\) in such a way that the resulting vertex is incident to all those edges (other than \(xy\)) which are originally incident to \(x\) or \(y\). We say that \(e\) is contractible if \(F\) is \(k\)-connected. A bowtie is the graph consisting of two triangles with exactly one vertex in common. We prove that if a \(k\)-connected graph \(G\) (\(k \geq 4\)) has no contractible edge, then there exists a bowtie in \(G\).

We prove that the number of nonisomorphic minimal \(2\)-colorings of the edges of \(K_{4n+3}\) is at least \(2n\) less than the number of nonisomorphic minimal \(2\)-colorings of the edges of \(K_{4n+2}\), where \(n\) is a nonnegative integer. Harary explicitly gave all the nonisomorphic minimal \(2\)-colorings of the edges of \(K_6\). In this paper, we give all the nonisomorphic minimal \(2\)-colorings of the edges of \(K_7\).

We restate a recent improvement of the inclusion-exclusion principle in terms of valuations on distributive lattices and present a completely new proof of the result. Moreover, we establish set-theoretic identities and logical equivalences of inclusion-exclusion type, which have not been considered before.

Let \(\delta(G)\) denote the minimum degree of a graph \(G\). We prove that for \(t \geq 4\) and \(k \geq 2\), a graph \(G\) of order at least \((t + 1)k + 2t^2 – 4t + 2\) with \(\delta(G) \geq k+t- 1\) contains \(k\) pairwise vertex-disjoint \(K_{1,t}\)’s.

In this paper, we construct a squag \(SQG(3n)\) of cardinality \(3n\) that contains three given arbitrary squags \(SQG(n)\)s as disjoint subquags. Accordingly, we can construct a subdirectly irreducible squag \(SQG(3n)\), for each \(n \geq 7\), with \(n \equiv 0, 3 \pmod{6}\). Also, we want to review the shape of the congruence lattice of non-simple squags \(SQG(n)\) for some \(n\) and to give a classification of the class of all \(SQG(21)\)s and the class of all \(SQG(27)\)s according to the shape of its congruence lattice. \(SQG(21)\)s are classified into three classes and \(SQG(27)\)s are classified into four classes. The construction of \(SQG(3n)\), which is given in this paper, helps us to construct examples of each class of both \(SQG(21)\)s and \(SQG(27)\)s.

We show how to produce algebraically a complete orthogonal set of Latin squares from a left quasifield and how to generate algebraically a maximal set of self-orthogonal Latin squares from a left nearfield.

A \((k;g)\)-graph is a \(k\)-regular graph with girth \(g\). A \((k; g)\)-cage is a \((k; g)\)-graph with the least possible number of vertices. In this paper, we prove that all \((4; g)\)-cages are \(4\)-connected, a special case of the conjecture about \((k; g)\)-cages’ connectivity made by H.L. Fu \(et\; al [1]\).