The spectrum problem for decomposition of trees with up to eight edges was introduced and solved in 1978 by Huang and Rosa. Additionally, the packing problem was settled for all trees with up to six edges by Roditty. For the first time, we consider obtaining all possible leaves in a maximum tree-packing of \(K_n\), which we refer to as the spectrum problem for packings of complete graphs. In particular, we completely solve this problem for trees with at most five edges. The packing designs are used in developing optimal error-correcting codes, which have applications in biology, such as in DNA sequencing.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A labeling \(f\) of a graph \(G\) is said to be edge-friendly if \(|e_f(0) – e_f(1)| \leq 1\), where \(e_f(i) = \text{card}\{e \in E(G) : f(e) = i\}\). An edge-friendly labeling \(f : E(G) \to \mathbb{Z}_2\) induces a partial vertex labeling \(f^+ : V(G) \to A\) defined by \(f^+(x) = 0\) if the edges incident to \(x\) are labeled \(0\) more than \(1\). Similarly, \(f^+(x) = 1\) if the edges incident to \(x\) are labeled \(1\) more than \(0\). \(f^+(x)\) is not defined if the edges incident to \(x\) are labeled \(1\) and \(0\) equally. The edge-balance index set of the graph \(G\), \(EBI(G)\), is defined as \(\{|v_f(0) – v_f(1)| : \text{the edge labeling } f \text{ is edge-friendly}\}\), where \(v_f(i) = \text{card}\{v \in V(G) : f^+(v) = i\}\).

An \(n\)-wheel is a graph consisting of \(n\) cycles, with each vertex of the cycles connected to one central hub vertex. The edge-balance index sets of \(n\)-wheels are presented.

A set of vertices \(W\) \emph{locally resolves} a graph \(G\) if every pair of adjacent vertices is uniquely determined by its coordinate of distances to the vertices in \(W\). The minimum cardinality of a local resolving set of \(G\) is called the \emph{local metric dimension} of \(G\). A graph \(G\) is called a \(k\)-regular graph if every vertex of \(G\) is adjacent to \(k\) other vertices of \(G\). In this paper, we determine the local metric dimension of an \((n-3)\)-regular graph \(G\) of order \(n\), where \(n \geq 5\).

Let \(\mathcal{G}_n\) be the set of all simple loopless undirected graphs on \(n\) vertices. Let \(T\) be a linear mapping, \(T: \mathcal{G}_n \to \mathcal{G}_n\), such that the dot product dimension of \(T(G)\) is the same as the dot product dimension of \(G\) for any \(G \in \mathcal{G}_n\). We show that \(T\) is necessarily a vertex permutation. Similar results are obtained for mappings that preserve sets of graphs with specified dot product dimensions.

A permutation \(\pi\) on a set of positive integers \(\{a_1, a_2, \ldots, a_n\}\) is said to be graphical if there exists a graph containing exactly \(a_i\) vertices of degree \(\pi(a_i)\) for each \(i\) (\(1 \leq i \leq n\)). It has been shown that for positive integers with \(a_1 < a_2 < \ldots < a_n\), if \(\pi(a_n) = a_n\), then the permutation \(\pi\) is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i)\).

We use a criterion of Tripathi and Vijay to provide a new proof of this result and to establish a similar result for permutations \(\pi\) such that \(\pi(a_{n-1}) = a_n\). We prove that such a permutation is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_na_{n-1} \leq a_{n-1}(a_{n-1} – 1) + \sum_{i\neq n-1} a_i\pi(a_i)\). We also consider permutations such that \(\pi(a_n) = a_{n-1}\) and, more generally, those such that \(\pi(a_n) = a_{n-j}\) for some \(j\) (\(1 < j < n\)).

A \(k\)-labeling of a graph is a labeling of the vertices of the graph by \(k\)-tuples of non-negative integers such that two vertices of \(G\) are adjacent if and only if their label \(k\)-tuples differ in each coordinate. The dimension of a graph \(G\) is the least \(k\) such that \(G\) has a \(k\)-labeling.

Lovász et al. showed that for \(n \geq 3\), the dimension of a path of length \(n\) is \((\log_2 n)^+\). Lovász et al. and Evans et al. determined the dimension of a cycle of length \(n\) for most values of \(n\). In the present paper, we obtain the dimension of a caterpillar or provide close bounds for it in various cases.