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.

In this paper we define new generalizations of Fibonacci numbers and Lucas numbers in the distance sense. These generalizations are closely related to the concept of \((2,k )\)-distance Fibonacci numbers presented in \([10]\). We show some applications of these numbers in number decompositions and we also define a new type of Lucas numbers.

For a vector \({R} = (r_1, r_2, \ldots, r_m)\) of non-negative integers, a mixed hypergraph \(\mathcal{H}\) is a realization of \({R}\) if its chromatic spectrum is \({R}\). In this paper, we determine the minimum number of vertices of realizations of a special kind of vectors \({R}_2\). As a result, we partially solve an open problem proposed by Král in \(2004\).