When \(G\) and \(F\) are graphs, \(v \in V(G)\) and \(\varphi\) is an orbit of \(V(F)\) under the action of the automorphism group of \(F\), \(s(F,G,v,\varphi)\) denotes the number of induced subgraphs of \(G\) isomorphic to \(F\) such that \(v\) lies in orbit \(\theta\) of \(F\). Vertices \(v \in V(G)\) and \(w \in V(H)\) are called \(k\)-vertex subgraph equivalent (\(k\)-SE), \(2 \leq k < n = |V(G)|\), if for each graph \(F\) with \(k\) vertices and for every orbit \(\varphi\) of \(F\), \(s(F,G,v,\varphi) = s(F,H,w,\varphi)\), and they are called similar if there is an isomorphism from \(G\) to \(H\) taking \(v\) to \(w\). We prove that \(k\)-SE vertices are \((k-1)\)-SE and several parameters of \((n-1)\)-SE vertices are equal. It is also proved that in many situations, “(n-1)-SE between vertices is equivalent to their similarity'' and it is true always if and only if Ulam's Graph Reconstruction Conjecture is true.

External Difference Families \((EDFs)\) are a new type of combinatorial designs originated from cryptography. In this paper, some constructions of \(EDFs\) are presented by using Gauss sums. Several classes of \(EDFs\) and related combinatorial designs are obtained.

The crossing number problem is in the forefront of topological graph theory. At present, there are only a few results concerning crossing numbers of join of some graphs. In this paper, for the special graph \(Q\) on six vertices, we give the crossing numbers of its join with \(n\) isolated vertices, as well as with the path \(P_n\) on \(n\) vertices and with the cycle \(C_n\).

In this article, we give a generalization of the multiparameter non-central Stirling numbers of the first and second kinds, Lah numbers, and harmonic numbers. Some new combinatorial identities, new explicit formulas, and many relations between different types of Stirling numbers and generalized harmonic numbers are found. Moreover, some interesting special cases of the generalized multiparameter non-central Stirling numbers are deduced. Furthermore, a matrix representation of the results obtained is given and a computer program is written using Maple and executed for calculating \(GMPNSN-1\) and their inverse \((GMPNSN-2)\), along with some of their interesting special cases.

Suppose \(G\) is a graph. Let \(u\) be a vertex of \(G\). A vertex \(v\) is called an \(i\)-neighbor of \(u\) if \(d_G(u,v) = i\). A \(1\)-neighbor of \(u\) is simply called a neighbor of \(u\). Let \(s\) and \(t\) be two nonnegative integers. Suppose \(f\) is an assignment of nonnegative integers to the vertices of \(G\). If the following three conditions are satisfied, then \(f\) is called an \((s, t)\)-relaxed \(L(2,1)\)-labeling of \(G\): (1) for any two adjacent vertices \(u\) and \(v\) of \(G\), \(f(u) \neq f(v)\); (2) for any vertex \(u\) of \(G\), there are at most \(s\) neighbors of \(u\) receiving labels from \(\{f(u) – 1, f(u)+ 1\}\); (3) for any vertex \(u\) of \(G\), the number of \(2\)-neighbors of \(u\) assigned the label \(f(u)\) is at most \(t\). The minimum span of \((s, t)\)-relaxed \(L(2,1)\)-labelings of \(G\) is called the \((s,t)\)-relaxed \(L(2,1)\)-labeling number of \(G\), denoted by \(\lambda_{2,1}^{s,t}(G)\). It is clear that \(\lambda_{2,1}^{0,0}(G)\) is the so-called \(L(2, 1)\)-labeling number of \(G\). In this paper, the \((s, t)\)-relaxed \(L(2, 1)\)-labeling number of the hexagonal lattice is determined for each pair of two nonnegative integers \(s\) and \(t\). And this provides a series of channel assignment schemes for the corresponding channel assignment problem on the hexagonal lattice.

As an additive weight version of the Harary index, the reciprocal degree distance of a simple connected graph \(G\) is defined as \(RDD(G) = \sum\limits_{u,v \subseteq V(G)} \frac{d_G(u)+d_G(v)}{d_G(u,v)}\), where \(d_G(u)\) is the degree of \(u\) and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). In this paper, we respectively characterize the extremal graphs with the maximum \(RDD\)-value among all the graphs of order \(n\) with given number of cut vertices and cut edges. In addition, an upper bound on the reciprocal degree distance in terms of the number of cut edges is provided.

In this work, linear codes over \(\mathbb{Z}_{2^s}\) are considered together with the extended Lee weight, which is defined as
\[w_L(a) = \begin{cases}
a & \text{if } a \leq 2^{s-1}, \\
2^s – x & \text{if } a > 2^{s-1}.
\end{cases}\]
The ideas used by Wilson and Yildiz are employed to obtain divisibility properties for sums involving binomial coefficients and the extended Lee weight. These results are then used to find bounds on the power of 2 that divides the number of codewords whose Lee weights fall in the same congruence class modulo \(2^e\). Comparisons are made with the results for the trivial code and the results for the homogeneous weight.

In this paper we study the Laplacian spectral radius of bicyclic graphs with given independence number and characterize the extremal graphs completely.

In this paper, we obtain some analytical expressions and give two simple formulae for the expected values of the Wiener indices of the random Phenylene and Spiro hexagonal chains.

Let \(G\) be a bicyclic graph. Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph with the maximal signless Laplacian spectral radius among all the bicyclic graphs with \(n\) vertices and diameter \(d\).