It may be desired to seat \(n\) people along a row (as at a lunch counter), or \(n+1\) people around a circular table, in \(n\) consecutive rounds of seating, so that each person \(x\) has every other person \(y\) on their right exactly once, and on their left exactly once, in one of the seatings. Alternatively, it may be desired to seat \(2n\) people along a row, or \(2n + 1\) people around a circular table, in only \(n\) consecutive rounds, so that each person \(x\) is adjacent to every other person \(y\) (either on the right or the left) exactly once. We show that these problems are solved using the rows of Tuscan squares to specify the successive rounds of seatings.

Let \(\mathbb{F}_q^(n+1)\) denote the \((n+l)\)-dimensional projective space over a finite field \(\mathbb{F}_q\). For a fixed integer \(m \leq \min\{n,l\}\), denote by \(\mathcal{L}_o^m(\mathbb{F}_q^{n+1})\) the set of all subspaces of type \((t,t_1)\), where \(t_1 \leq t \leq m\). Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute all its parameters.

If \(X\) is a geodesic metric space and \(x_1, x_2, x_3 \in X\), a geodesic triangle \(T = \{x_1, x_2, x_3\}\) is the union of the three geodesics \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) in \(X\). The space \(X\) is \(\delta\)-hyperbolic (in the Gromov sense) if any side of \(T\) is contained in a \(5\delta\)-neighborhood of the union of the two other sides, for every geodesic triangle \(T\) in \(X\). We denote by \(\delta(X)\) the sharp hyperbolicity constant of \(X\), i.e., \(\delta(X) := \inf\{\delta \geq 0: X \text{ is } \delta\text{-hyperbolic}\}\). The main result of this paper is the inequality \(\delta(G) \leq \delta(\mathcal{L}(G))\) for the line graph \(\mathcal{L}(G)\) of every graph \(G\). We prove also the upper bound \(\delta(L(G)) \leq 5\delta(G) + 3l_{\max}\), where \(\max\) is the supremum of the lengths of the edges of \(G\). Furthermore, if every edge of \(G\) has length \(k\), we obtain \(\delta(G) \leq \delta(\mathcal{L}(G)) \leq 5\delta(G) + 5k/2\).

For graphs \(G\) and \(H\), the size-balanced Ramsey multipartite number \(m_j(G, H)\) is defined as the smallest positive integer \(s\) such that any arbitrary red/blue coloring of the graph \(K_{s,s}\) forces the appearance of a red \(G\) or a blue \(H\). In the main case of this paper, we generalize methods used in finding bipartite Ramsey numbers for \(b(nK_2, mK_2)\) to finding the balanced Ramsey multipartite number \(m_j(nK_2, mK_2)\).

The subdivision graph \(S(G)\) of a graph \(G\) is the graph obtained by inserting a new vertex into every edge of \(G\). Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs. The subdivision-vertex corona of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(|V(G_1)|\) copies of \(G_2\), all vertex-disjoint, by joining the \(i\)th vertex of \(V(G_1)\) to every vertex in the \(i\)th copy of \(G_2\). The subdivision-edge corona of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus G_2\), is the graph obtained from \(S(G_1)\) and \(|I(G_1)|\) copies of \(G_2\), all vertex-disjoint, by joining the \(i\)th vertex of \(I(G_1)\) to every vertex in the \(i\)th copy of \(G_2\), where \(I(G_1)\) is the set of inserted vertices of \(S(G_1)\). In this paper, we determine the generalized characteristic polynomial of \(G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)). As applications, the results on the spectra of \( G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)) enable us to construct infinitely many pairs of \(\Phi\)-cospectral graphs. The adjacency spectra of \(G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)) help us to construct many infinite families of integral graphs. By using the Laplacian spectra, we also obtain the number of spanning trees and Kirchhoff index of \(G_1 \odot G_2\) and \(G_1 \ominus G_2\), respectively.

In this paper, we study arc-transitive pentavalent graphs of order \(4p^n\), where \(p\) is a prime and \(n\) is a positive integer. It is proved that no such graph exists for each prime \(p \geq 5\), and all such graphs with \(p = 2\) or \(3\) which are \(G\)-basic (that is, \(G\) has no non-trivial normal subgroup such that the graph is a normal cover of the corresponding normal quotient graph) are determined. Moreover, as an application, arc-transitive pentavalent graphs of order \(4p^2\) and \(4p^3\) are determined.

In \(1982\), Beutelspacher and Brestovansky determined the \(2\)-color Rado number of the equation \[x_1+ x_2 x + \ldots +x_{m-1} =x_{ m} \] for all \(m \geq 3\). Here we extend their result by determining the 2-color Rado number of the equation \[x_1 +x_2 + \dots + x_n = y_1 +y_2+ \ldots + y_k\] for all \(n \geq 2\) and \(k \geq 2\). As a consequence, we determine the 2-color Rado number of \[x_1+ x_2 + \ldots + x_n = a_1 y_1 + \dots + a_\ell y_\ell\] in all cases where \(n \geq 2\) and \(n \geq a_1 + \dots + a_\ell\), and in most cases where \(n \geq 2\) and \(2n \geq a_1 + \dots + a_\ell\).

The subdivision graph \(S(G)\) of a graph \(G\) is the graph obtained by inserting a new vertex into every edge of \(G\). The set of inserted vertices of \(S(G)\) is denoted by \(I(G)\). Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs. The subdivision-edge-vertex join of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(I(G_1)\) to every vertex in \(V(G_2)\). The subdivision-edge-edge join of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(I(G_1)\) to every vertex in \(I(G_2)\). The subdivision-vertex-edge join of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(V(G_1)\) to every vertex in \(I(G_2)\). In this paper, we obtain the formulas for resistance distance of \(G_1 \odot G_2\), \(G_1 \ominus G_2\), and \(G_1 \odot G_2\).

A hypergraph is intersecting if any two different edges have exactly one common vertex, and an \(n\)-quasicluster is an intersecting hypergraph with \(n\) edges, each one containing at most \(n\) vertices, and every vertex is contained in at least two edges. The Erdős-Faber-Lovász Conjecture states that the chromatic number of any \(n\)-quasicluster is at most \(n\). In the present note, we prove the correctness of the conjecture for a new infinite class of \(n\)-quasiclusters using a specific edge coloring of the complete graph.

Let \(G\) be a graph of order \(n\) and let \(\Phi(G, \lambda) = \det(\lambda I_n – L(G)) = \sum_{k=0}^{n}(-1)^k c_k(G) \lambda^{n-k}\) be the characteristic polynomial of the Laplacian matrix of a graph \(G\). In this paper, we identify the minimal Laplacian coefficients of unicyclic graphs with \(n\) vertices and diameter \(d\). Finally, we characterize the graphs with the smallest and the second smallest Laplacian-like energy among the unicyclic graphs with \(n\) vertices and fixed diameter \(d\).