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 \(d\).

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of the weights \(\frac{2}{d_u+ d_v}\) of all edges \(uv\) of \(G\), where \(d_u\) denotes the degree of a vertex \(u\) in \(G\). We determine the \(n\)-vertex trees with the second and third maximum harmonic indices for \(n \geq 7\), the fourth maximum harmonic index for \(n \geq 10\), and fifth maximum harmonic index for $n \geq 11\), and unicyclic graphs with the second and third maximum harmonic indices for \(n \geq 5\), the fourth maximum harmonic index for \(n \geq 7\), and fifth maximum harmonic index for \(n \geq 8\), and bicyclic graphs with the maximum harmonic index for \(n \geq 6\), the second and third maximum harmonic indices for \(n \geq 7\), and fourth maximum harmonic index for \(n \geq 9\).

Graph embedding has been known as a powerful tool for implementation of parallel algorithms and simulation of different interconnection networks. In this paper, we obtain the minimum wirelength of embedding circulant networks into necklace and windmill graphs. The algorithms for obtaining the same are of \(O(2n)\)-linear time.

In this paper, a reliable symbolic computational algorithm is presented for inverting a general companion matrix by using parallel computing along with recursion. The computational cost of the algorithm is \(O(n^2)\). The algorithm is implementable to the Computer Algebra System (CAS) such as MAPLE, MATLAB, and MATHEMATICA. Three examples are presented for the sake of illustration.

Let \(K_r\) be the complete graph on \(r\) vertices in which there exists an edge between every pair of vertices, \(K_{m,n}\) be the complete bipartite graph with \(m\) vertices in one partition and \(n\) vertices in the other partition, where each vertex in one partition is adjacent to each vertex in the other partition, and \(K(n, r)\) be the complete \(r\)-partite graph \(K_{n,n,…,n}\) where each partition has \(n\) vertices. In this paper, we determine the minimum number of monochromatic stars \(K_{1,p}\), \( \forall p \geq 2\), in any \(t\)-coloring (\(t \geq 2\)) of edges of \(K_r\), \(K_{m,n}\), and \(K(n, r)\). Also, we prove that these lower bounds are sharp for all values of \(m, n, p, r\), and \(t\) by giving explicit constructions.

In this paper, we prove that if the toughness of a \(k\)-tree \(G\) is at least \(\frac{k+1}{3}\), then \(G\) is panconnected for \(k \geq 3\), or \(G\) is vertex pancyclic for \(k = 2\). This result improves a result of Broersma, Xiong, and Yoshimoto.

Since the Wiener index has been successful in the study of benzenoid systems and boiling points of alkanes, it is natural to examine this number for the study of fullerenes, most of whose cycles are hexagons. This topological index is equal to the sum of distances between all pairs of vertices of the respective graph. It was introduced in \(1947\) by one of the pioneers of this area, Harold Wiener, who realized that there are correlations between the boiling points of paraffins and the structure of the molecules. The present paper is the first attempt to compute the Wiener index of an infinite class of fullerenes. Further, we obtain a correlation between the values of the Wiener index and the boiling point of such fullerenes for the first time.