We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with \(C_{3}\) and \(C_{4}\) vanes, and infinite families of \(3,5\)-windmills and \(3,6\)-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of \(t\) 5-cycles with one vertex in common (\(C_{5}^{t}\)) is graceful if and only if \(t \equiv 0, 3 \pmod{4}\) and near graceful when \(t \equiv 1, 2 \pmod{4}\).

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence \((G_n)_{n \geq 0}\), where \(G_n\) is the group generated by a subset of the symmetric group \(S_n\) consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence \((G_n)_{n \geq 0}\) is eventually constant. Moreover, we find a set of patterns such that \((G_n)_{n \geq 0}\) is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.

We consider the eccentric graph of a graph \(G\), denoted by \(\mathrm{ecc}(G)\), which has the same vertex set as \(G\), and two vertices in the eccentric graph are adjacent if and only if their distance in \(G\) is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between \(\mathrm{ecc}(G)\) and the complement of \(G\), and show that the previous necessary condition given in the literature is inadequate. Also, we obtain that the diameter of \(\mathrm{ecc}(T)\) is at most 3 for any tree and get some characterizations of the eccentric graph of trees.

Let \(G\) be a finite simple undirected \((p, q)\)-graph, with vertex set \(V(G)\) and edge set \(E(G)\) such that \(p = |V(G)|\) and \(q = |E(G)|\). A super edge-magic total labeling \(f\) of \(G\) is a bijection \(f \colon V(G) \cup E(G) \longrightarrow \{1, 2, \dots, p+q\}\) such that for all edges \(uv \in E(G)\), \(f(u) + f(v) + f(uv) = c(f)\), where \(c(f)\) is called a magic constant, and \(f(V(G)) = \{1, \dots, p\}\). The minimum of all \(c(f)\), where the minimum is taken over all the super edge-magic total labelings \(f\) of \(G\), is defined to be the super edge-magic total strength of the graph \(G\). In this article, we work on certain classes of unicyclic graphs and provide evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \((p, q)\)-graphs is equal to \(2q + \frac{n+3}{2}\).

For a poset \(P = C_a \times C_b\), a subset \(A \subseteq P\) is called a chain blocker for \(P\) if \(A\) is inclusion-wise minimal with the property that every maximal chain in \(P\) contains at least one element of \(A\), where \(C_i\) is the chain \(1 < \cdots < i\). In this article, we define the shelter of the poset \(P\) to give a complete description of all chain blockers of \(C_5 \times C_b\) for \(b \geq 1\).

This project aims at investigating properties of channel detecting codes on specific domains \(1^+0^+\). We focus on the transmission channel with deletion errors. Firstly, we discuss properties of channels with deletion errors. We propose a certain kind of code that is a channel detecting (abbr. \(\gamma\)-detecting) code for the channel \(\gamma = \delta(m, N)\) where \(m < N\). The characteristic of this \(\gamma\)-detecting code is considered. One method is provided to construct \(\gamma\)-detecting code. Finally, we also study a kind of special channel code named \(\tau(m, N)\)-srp code.

A chemical structure specifies the molecular geometry of a given molecule or solid in the form of atom arrangements. One way to analyze its properties is to simulate its formation as a product of two or more simpler graphs. In this article, we take this idea to find upper and lower bounds for the generalized Randić index \(\mathcal{R}_{\alpha}\) of four types of graph products, using combinatorial inequalities. We finish this paper by providing the bounds for \(\mathcal{R}_{\alpha}\) of a line graph and rooted product of graphs.

Let \(G\) be a \((p, q)\) graph. Let \(f: V(G) \to \{1, 2, \ldots, k\}\) be a map where \(k \in \mathbb{N}\) is a variable and \(k > 1\). For each edge \(uv\), assign the label \(\gcd(f(u), f(v))\). \(f\) is called \(k\)-Total prime cordial labeling of \(G\) if \(\left|t_{f}(i) – t_{f}(j)\right| \leq 1\), \(i, j \in \{1, 2, \ldots, k\}\) where \(t_{f}(x)\) denotes the total number of vertices and edges labeled with \(x\). A graph with a \(k\)-total prime cordial labeling is called \(k\)-total prime cordial graph. In this paper, we investigate the 4-total prime cordial labeling of some graphs like dragon, Möbius ladder, and corona of some graphs.

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An edge labeling \(f: E \to Z_{2}\) induces a vertex labeling \(f^{+} : V \to Z_{2}\) defined by \( f^{+}(v) \equiv \sum_{uv \in E} f(uv) \pmod 2 \), for each vertex \(v \in V\). For \(i \in Z_{2}\), let \( v_{f}(i) = |\{v \in V : f^+(v) = i\}| \) and \( e_{f}(i) = |\{e \in E : f(e) = i\}| \). An edge labeling \(f\) of a graph \(G\) is said to be edge-friendly if \( |e_{f}(1) – e_{f}(0)| \le 1 \). The set \(\{v_f(1) – v_f(0) : f \text{ is an edge-friendly labeling of } G\}\) is called the full edge-friendly index set of \(G\). In this paper, we shall determine the full edge-friendly index sets of one point union of cycles.