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 \(SQ(21)s\).

The well-known Petersen graph \(G(5,2)\) admits drawings in the ordinary Euclidean plane in such a way that each edge is represented as a line segment of length \(1\). When two vertices are drawn as the same point in the Euclidean plane, drawings are said to be degenerate. In this paper, we investigate all such degenerate drawings of the Petersen graph and various relationships among them. A heavily degenerate unit distance planar representation, where the representation of a vertex lies in the interior of the representation of an edge it does not belong to, is also shown.

The distance spectral radius of a connected graph \(G\), denoted by \(\rho(G)\), is the maximal eigenvalue of the distance matrix of \(G\). In this paper, we find a sharp lower bound as well as a sharp upper bound of \(\rho(G)\) in terms of \(\omega(G)\), the clique number of \(G\). Furthermore, both extremal graphs are uniquely determined.

Let \(G\) be a graph with \(n\) vertices. The vertex matching polynomial \(M_v(G, x)\) of the graph \(G\) is defined as the sum of \((-1)^rq_v(G,r)x^{n-r}\), in which \(q_v(G,r)\) is the number of \(r\)-vertex independent sets. In this paper, we extend some important properties of the matching polynomial to the vertex matching polynomial \(M_v(G,2x)\). The matching and vertex matching polynomials of some important class of graphs and some applications in nanostructures are presented.

In \([18]\), Farrell and Whitehead investigate circulant graphs that are uniquely characterized by their matching and chromatic polynomials (i.e., graphs that are “matching unique” and “chromatic unique”). They develop a partial classification theorem, by finding all matching unique and chromatic unique circulants on \(n\) vertices, for each \(n \leq 8\). In this paper, we explore circulant graphs that are uniquely characterized by their independence polynomials. We obtain a full classification theorem by proving that a circulant is independence unique if and only if it is the disjoint union of isomorphic complete graphs.

We present a formula for the number of line segments connecting \(q+1\) points of an \(n_1 \times \cdots \times n_k\) rectangular grid. As corollaries, we obtain formulas for the number of lines through at least \(k\) points and, respectively, through exactly \(k\) points of the grid. The well-known case \(k = 2\) is thus generalized. We also present recursive formulas for these numbers assuming \(k = 2, n_1 = n_2\). The well-known case \(q = 2\) is thus generalized.

Let \(H\) and \(G\) be two graphs, where \(G\) is a simple subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \((H,G)\)-GD, is a partition of all the edges of \(H\) into subgraphs (\(G\)-blocks), each of which is isomorphic to \(G\). A large set of \((H, G)\)-GD, denoted by \((H,G)\)-LGD, is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \((H,G)\)-GDs. In this paper, we determine the existence spectrums for \((\lambda K_{m,n}, P_3)\)-EGD and \((\lambda K_{n,n,n}, P_3)\)-LGD.

The support of a \(t\)-design is the set of all distinct blocks in the design. The notation \(t-(v,k, \lambda|b^*)\) is used to denote a \(t\)-design with precisely \(b^*\) distinct blocks. We present some results about the structure of support in \(t\)-designs. Some of them are about the number and the range of occurrences of \(i\)-sets (\(1 \leq i \leq t\)) in the support. A new bound for the support sizes of \(t\)-designs is presented. In particular, given a \(t-(v, k, \lambda|b^*)\) design with \(b > b_0\), where \(b\) and \(b_0\) are the cardinality and the minimum cardinality of block sets in the design, respectively, then it is shown that \(b^* \geq \lceil \frac{\lceil \frac{2b}{\lambda}\rceil +7}{2}\rceil\). We also show that when \(\lambda\) varies over all positive integers, then there is no \(t-(v,k,\lambda | b^*)\)-design with the support sizes equal to \(b^*_{min}+1, b^*_{min}+2\) and \(b^*_{min}+3\), where \(b^*_{min}\) denotes the least possible cardinality of the support sizes in this design.