A prime labeling of a simple finite graph \(G\) is a labeling of the vertices with distinct integers from \(\{1,2, \dots, |G|\}\) such that the labels of any two adjacent vertices are coprime. If \(G\) admits such a labeling, we call \(G\) a prime graph. If the set \(\{1,2, \dots, |G|\}\) is replaced by \(\{k,k+1, \dots, k+|G|-1\}\) in the previous definition, then we call it \(k\)-prime labeling, where \(k\) is a positive integer. Since some graph might be 2-prime labeling but not 3-prime labeling, to avoid ambiguity, we define uniform prime labeling for graphs. Focused on trees, using some very classical number theory results, we show that any star is uniform prime if and only if it has at most 16 vertices. We also prove any tree of order \(n\) with diameter \(n-1\), \(n-2\), \(n-3\), and \(n-4\) are uniform prime. Based on these results and several trees with diameter \(n-5\), we show that any tree with at most 8 vertices (There are 1+1+1+2+3+6+11+23=48 such non-isomorphic trees) are uniform prime. We also verified the uniform primality of several periodically constructed trees. We post some open questions and conjectures which should be essential and interesting by the end of Section 4. For example, we conjecture all the trees with order up to 16 are uniform prime and ask if there exists a tree other than star is not uniform prime.

Let \(G(V,E)\) be a finite simple graph. Denote by \(|V|\) and \(|E|\) the cardinality of the set \(V\) and \(E\), respectively. An \(\alpha\)-labeling \(f\) is an injective function \({f}:V\rightarrow \{0,1,2,…, |E|\}\) such that \(\{|f(u)-f(v)|:uv\in E\}=\{1,2,…,|E|\}\) and for a constant \(\lambda\), \(f(u)\le \lambda <f(v)\) for every \(uv\in E\). A graph that has an \(\alpha\)-labeling is called \(\alpha\)-graph. An open chain graph, or shortly a chain graph, is a graph with blocks \({B_1,B_2,…,B_n}\) such that for every \(i\), \(B_i\) and \(B_{i+1}\) have a common vertex, in such a way that the block-cut-vertex graph is a path. A chain graph having \({n}\) blocks \({B_1,B_2,…,B_n}\) is denoted by \([B_{1},B_{2},…,B_{n}]\). Let \({c_i}\) be the common vertex of \({B_i}\) and \({B_{i+1}}\), in \([{B_1,B_2,…,B_n}]\), \(1\le i \le n-1\). Consider a vertex of \({B_1}\), \({c_0 \ne c_1}\), and a vertex of \({B_n}\), \({c_n \ne c_{n-1}}\). Assume that \(B\) is another block graph and \(x\) and \(y\) are two different vertices in \(B\). The graph which is constructed by identifying \(c_0\) with \(x\) and \(c_n\) with \(y\) is called closed chain graph, and is denoted by \([{B_1,B_2,…,B_n,B}]_c\). By using pattern recognition and axiomatic deductive method we get results that some chain graphs with blocks of complete bipartite graphs are \(\alpha\)-labeling.

Threshold graphs are graphs whose node set can be partitioned into a clique and an independent set, with the additional property that for each pair of nodes, one’s neighborhood is a subset of the other’s neighborhood. Threshold graphs have been well-studied in graph theory, but not much is known about multigraphs that are underlying threshold. Proper threshold graphs are those in which all nodes in the independent set have the same degree. In this paper, we present a formula for the eigenvalues of a particular class of multigraphs that are underlying proper threshold.

Recently, the notion of a Weyl substructure in a spherical building was introduced type by type. In this paper we provide a uniform (axiomatic) definition across all types. In particular, this provides a new characterisation of the Ree-Tits octagons. We then show that uniclass automorphisms of spherical buildings are uniformly characterised by their fix structure. For type preserving automorphisms, this follows from earlier work, and so the focus here is on dualities. In particular, it follows that a duality pointwise fixing a Weyl substructure is automatically a polarity. This characterises all polarities in self-dual spherical buildings where opposition acts trivially on the types.