A connected graph \(G = (V, E)\) is \((a, d)\)-antimagic if there exist positive integers \(a, d\) and a bijection \(g: E \to \{1, 2, \ldots, |E|\}\) such that the induced mapping
\[f_g = \Sigma\{g(u,v): (u, v) \in E(G)\}\, \text{is injective and}\]
\[f_g(V) = \{a, a+d, a+2d, \ldots, a+(|V|-1)d\}.\]
In this paper, we prove two conjectures of Baca concerning \((a, d)\)-antimagic labelings of antiprisms

Some special sum graphs and difference graphs, based on abelian groups, are discussed. In addition to Li’s result on character sum estimates, Weil’s character sum estimates are also used to show that these are indeed Ramanujan graphs.

A critical set in a Latin square of order \(n\) is a set of entries in a Latin square which can be embedded in precisely one Latin square of order \(n\). Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order \(n\). A smallest critical set in a Latin square is a critical set of minimum cardinality. In this paper we find smallest critical sets for all the Latin squares of orders six and seven. We also find smallest critical sets of orders six and seven which are also weak critical sets. In particular, we find a weak critical set of size twelve for the dihedral group of order six.

We study combinatorial structure of \(\ell\)-optimal \(A^2\)-codes that offer the best protection for spoofing of order up to \(\ell\) and require the least number of keys for the transmitter and the receiver. We prove that for such codes the transmitter’s encoding matrix is a strong partially balanced resolvable design, and the receiver’s verification matrix corresponds to an \(\alpha\)-resolvable design with special properties.

It is proved in this paper that for any integer \(n \geq 136\), a SODLS(\(v, n\)) (self-orthogonal diagonal Latin square with missing subsquare) exists if and only if \(v \geq 3n+2\) and \(v-n\) even.

Employing trading signed design algorithm, we construct an automorphism-free \(4\)-\((15, 5, 5)\) design.

Consider those graphs \(G\) of size \(2n\) that have an eigenvalue \(\lambda\) of multiplicity \(n\) and where the edges between the star set and its complement is a matching. We show that \(\lambda\) must be either \(0\) or \(1\) and completely characterize the corresponding graphs.

We enumerate the 2-\((9,4,6)\) designs and find 270,474,142 non-isomorphic such designs in a backtrack search. The sizes of their automorphism groups vary between 1 and 360. Out of these designs, 19,489,464 are simple and 2,148,676 are decomposable.

A \(t\)-partite number is a \(t\)-tuple \(\vec{n} = (n_1, \ldots, n_t)\), where \(n_1, \ldots, n_t\) are positive integers. For a \(t\)-partite number \(\vec{n}\), let \(f_t(\vec{n})\) be the number of different ways to write \(\vec{n}\) as a product of \(t\)-partite numbers, where the multiplication is performed coordinate-wise, \((1, 1, \ldots, 1)\) is not used as a factor of \(\vec{n}\), and two factorizations are considered the same if they differ only in the order of the factors. This paper gives the following explicit upper bound for the multiplicative partition function \(f_t(\vec{n})\):
\[f_t(n_1, \ldots, n_t) \leq M^{w(t)},\, \text{where}\,\, M = \Pi_{i=1}^t n_i \,\,\text{and}\,\, w(t) = \frac{\log((t+1)1)}{t\log2}\].

The following partition problem was first introduced by R.C. Entringer and has subsequently been studied by the first author and more recently by Bollobas and Scott, who consider the hypergraph version as well, using a probabilistic technique. The partition problem is that of coloring the vertex set of a graph with \(s\) colors so that the number of induced edges is bounded for each color class. The techniques employed are non-constructive and non-probabilistic and improve the known bounds in the previous papers.

In a communication network, several vulnerability measures are used to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the edge-integrity of a graph is one \textbf{measure of graph vulnerability} and it is defined to be the minimum sum of the orders of a set of edges being removed and a largest remaining component. In this paper, the edge-integrity of graphs \(B_n\), \(H_n\), and \(E_p^t\), are calculated. Also, some results are given about edge-integrity of these graphs.

In this paper, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5, type \(h^n\) and index \(\lambda\), namely, \(n \geq 5\) and \(\lambda n(n-1)h^2 \equiv 0 \pmod{5}\), is also sufficient for \(\lambda \geq 2\). The result guarantees the analogous existence result for group divisible designs (GDDs) of type \(h^n\) having block size 5 and index \(4\lambda\).

The computational complexity of the graph isomorphism problem is still unknown. We consider Cartesian products \(K_n \times K_m\) of two complete graphs \(K_n\) and \(K_m\). An acyclic orientation of such a Cartesian product is called a sequence graph because it has an application in production scheduling. It can be shown that the graph isomorphism problem on the class of these acyclic digraphs is solvable in polynomial time. We give numbers of non-isomorphic sequence graphs for small \(n\) and \(m\). The orientation on the cliques of a sequence graph can be interpreted as job orders and machine orders of a shop scheduling problem with a complete operation set.

Tenacity is a recently introduced parameter to measure vulnerability of networks and graphs. We characterize graphs having the maximum number of edges among all graphs with a given number of vertices and tenacity.

In this paper, we show that some graphs are circuit unique by applying a new tool, which is the character of the matching polynomial. Some properties of the character of the matching polynomial is also given.

The theory of hypergeometric functions is brought to bear on a problem—namely, that of obtaining a certain power series expansion involving the sine function that is inclusive of the Catalan sequence and which serves as a prelude to the calculation of other related series of similar type. A general formulation provides the particular result of interest as a special case, into which Catalan numbers are introduced as desired.

A splitting partition for a graph \(G = (V, E)\) is a partition of \(V\) into sets \(R\), \(B\), and \(U\) so that the subgraphs induced by \(V – R\) and \(V – B\) are isomorphic. The splitting number \(\mu(G)\) is the size of \(|R|\) for any splitting partition which maximizes \(|R|\). This paper determines \(\mu(G)\) for trees of maximum degree at most three and exactly one degree two vertex and for trees all of whose vertices have degree three or one.

A decomposition of a digraph is said to be bicyclic if it admits an automorphism consisting of exactly two disjoint cycles. Necessary and sufficient conditions are given for the existence of bicyclic decompositions of the complete digraph into each of the four orientations of a 4-cycle.

The integrity of a graph \(G\), \(I(G)\), is defined by \(I(G) = min_{S \subseteq V(G)}\{|S| + m(G – S)\}\) where \(m(G – S)\) is the maximum order of the components of \(G – S\). In general, the integrity of an \(r\)-regular graph is not known [8]. We answer the following question for special regular graphs. For any given two integers \(p\) and \(r\) such that \(\frac{pr}{2}\) is an integer, is there an \(r\)-regular graph, say \(G^*\), on \(p\) vertices having size \(q = \frac{pr}{2}\) such that
\[I(G(p,\frac{pr}{2})) \leq I(G^*)\]
for all \(p\) and \(r\)? The \emph{integrity graph} is denoted by \(IG(p,n)\). It is a graph with \(p\) vertices, integrity \(n\), and has the least number of edges denoted by \(q[p,n]\). We compute \(q[p,n]\) for some values of \(p,n\).

If the distance between two vertices \(u\) and \(v\) in a graph \(G\) is \(k\), then \(u\) and \(v\) are said to \(k\)-step dominate each other. A set \(S\) of vertices of \(G\) is a \(k\)-step dominating set if every vertex of \(G\) is \(k\)-step dominated by some vertex of \(S\). The minimum cardinality of a \(k\)-step dominating set is the \(k\)-step domination number \(\rho_k(G)\) of \(G\). A sequence \(s: \ell_1, \ell_2, \ldots, \ell_k\) of positive integers is called an orbital dominating sequence for \(G\) if there exist distinct vertices \(v_1, v_2, \ldots, v_k\) of \(G\) such that every vertex of \(G\) is \(\ell_i\)-step dominated by \(v_i\) for some \(i\) (\(1 \leq i \leq k\)). An orbital dominating sequence \(s$ is minimal if no proper subsequence of \(s\) is an orbital dominating sequence for \(G\). The minimum length of a minimal orbital dominating sequence is the orbital domination number \(\gamma_{o}(G)\), while the maximum length of such a sequence is the upper orbital domination number \(\Gamma_{o}(G)\) of \(G\).

It is shown that for every pair \(i, j\) of positive integers with  \(i < j\), there exist graphs \(G\) and \(H\) such that both \(\rho_i(G) – \rho_j(G)\) and \(\rho_j(H) – \rho_i(H)\) are arbitrarily large. Also, there exist graphs \(G\) of arbitrarily large radius such that \(\gamma_{o}(G) < \rho_i(G)\) for every integer \(i\) (\(1 \leq i \leq \text{rad} G\)). All trees \(T\) with \(\gamma_{o}(T) = 3\) are characterized, as are all minimum orbital sequences of length 3 for graphs. All graphs \(G\) with \(\Gamma_{o}(G) = 2\) are characterized, as are all trees \(T\) with \(\Gamma_{o}(T) = 3\).