In the book embedding of an ordered set, the elements of the set are embedded along the spine of a book to form a linear extension. The pagenumber (or stack number) is the minimum number of pages needed to draw the edges as simple curves such that
edges drawn on the same page do not intersect. The pagenumber problem for ordered sets is known to be NP-complete, even if the order of the elements on the spine is-fixed. In this paper, we investigate this problem for some classes of ordered sets. We provide an efficient algorithm for embedding bipartite interval orders in a book with the minimum number of pages. We also give an upper bound for the pagenumber of general bipartite ordered sets and the pagenumber of complete multipartite ordered sets. At the end of this paper we discuss the effect of a number of diagram operations on the pagenumber of ordered sets. We give an answer to an open question by Nowakowski and Parker \([7]\) and we provide several known and new open questions we consider worth investigating.

Let \(\Gamma\) be a \(d\)-bounded distance-regular graph with diameter \(d \geq 2\).In this paper, we give some counting formulas of subspaces in \(\Gamma\) and construct an authentication code with perfect. secrecy.

We determine the full friendly index sets of spiders and disprove a conjecture by Lee and Salehi \([4]\) that the friendly index set of a tree forms an arithmetic progression.

Let \(k\) be a positive integer and \(G = (V(G), E(G))\) a graph. A subset \(S \subseteq V(G)\) is a \(k\)-dominating set if every vertex of \(V(G)- S\) is adjacent to at least \(k\) vertices of \(S\). The \(k\)-domination number \(\gamma_k(G)\) is the minimum cardinality of a \(k\)-dominating set of \(G\). A graph \(G\) is called \(\gamma_k\)-stable if \(\gamma_{\bar{k}}(G – e) = \gamma_{{k}}(G)\) for every edge \(e\) of \(E(G)\). We first provide a necessary and sufficient condition for \(\gamma_{\bar{k}}\)-stable graphs. Then, for \(k \geq 2\), we offer a constructive characterization of \(\gamma_{\bar{k}}\)-stable trees.

The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices joined by an edge if their product in the semigroup is zero. In this paper, we provide formulas to calculate the numbers of non-isomorphic zero-divisor semigroups corresponding to star graphs \(K_{1,m}\), two-star graphs \(T_{m,n}\), and windmill graphs, respectively.

Multisender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, a new multisender authentication codes with simultaneous model is constructed base on singular symplectic geometry over finite fields. The parameters and the maximum probabilities of deceptions are also computed.

Let \(D = (V, A)\) be a digraph with vertex set \(V\) and arc set \(A\). An absorbant of \(D\) is a set \(S \subseteq V\) such that for each \(v \in V \setminus S\), \(O(v) \cap S \neq \emptyset\), where \(O(v)\) is the out-neighborhood of \(v\). The absorbant number of \(D\), denoted by \(\gamma_a(D)\), is defined as the minimum cardinality of an absorbant of \(D\). The generalized de Bruijn digraph \(G_B(n, d)\) is a digraph with vertex set \(V(G_B(n, d)) = \{0, 1, 2, \ldots, n-1\}\) and arc set \(A(G_B(n, d)) = \{(x, y) \mid y = dx + i \, (\text{mod} \, n), 0 \leq i < d\}\). In this paper, we determine \(\gamma_a(G_B(n, d))\) for all \(d \leq n \leq 4d\).

We provide a concise combinatorial proof for the solution of the general two-term recurrence \(u(n, k) = u(n-1, k-1) + (a_{n-1}+b_{k})u(n-1, k)\), initially discovered by Mansour et al. \([4]\).

The vulnerability value of a communication network is the resistance of this communication network until some certain stations or communication links between these stations are disrupted and, thus communication interrupts. A communication network is modeled by a graph to measure the vulnerability as stations corresponding to the vertices and communication links corresponding to the edges, There are several types of vulnerability parameters depending upon the distance for each pair of two vertices. In this paper. closeness, vertex residual closeness (\(VRC\)) and normalized vertex residual closeness (\(NV RC\)) of some Mycielski graphs are calculated, furthermore upper and lower bounds are obtained.

A graph \(G\) is an {\([s, t]\)-graph if every subgraph induced by \(s\) vertices of \(G\) has at least \(t\) edges. This concept extends the independent number. In this paper, we prove that:

(1) if \(G\) is a \(k\)-connected \([k+2, 2]\)-graph, then \(G\) has a Hamilton cycle or \(G\) is isomorphic to the Petersen graph or \(\overline{K_{k+1}} \vee G_k\),

(2) if \(G\) is a \(k\)-connected \([k+3, 2]\)-graph, then \(G\) has a Hamilton path or \(G\) is isomorphic to \(\overline{K_{k+1}} \vee G_k\),
where \(G_r\) is an arbitrary graph of order \(k\). These two results generalize the following known results obtained by Chvátal-Erdős and Bondy, respectively:

(a) if \(\alpha(G)\leq \kappa(G) \) of order \(n \geq 3\), then \(G\) has a Hamilton cycle,

(b) if \(\alpha(G) – 1 \leq \kappa(G)\) , then \(G\) has a Hamilton path.