For two vertices \(u\) and \(v\) in a strong oriented graph \(D\), the strong distance \(\operatorname{sd}(u,v)\) between \(u\) and \(v\) is the minimum size (the number of arcs) of a strong sub-digraph of \(D\) containing \(u\) and \(v\). For a vertex \(v\) of \(D\), the strong eccentricity \(\operatorname{se}(v)\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong radius \(\operatorname{srad}(D)\) is the minimum strong eccentricity among the vertices of \(D\). The strong diameter \(\operatorname{sdiam}(D)\) is the maximum strong eccentricity among the vertices of \(D\). In this paper, we investigate the strong distances in strong oriented complete \(k\)-partite graphs. For any integers \(\delta, r, d\) with \(0 \leq \delta \leq \lceil\frac{k}{2}\rceil, 3 \leq r \leq \lfloor\frac{k}{2}\rfloor, 4 \leq d \leq k\), we have shown that there are strong oriented complete \(k\)-partite graphs \(K’, K”, K”’\) such that \(\operatorname{sdiam}(K’) – \operatorname{srad}(K’) = \delta, \operatorname{srad}(K”) = r\), and \(\operatorname{sdiam}(K”’) = d\).

The \(t\)-pebbling number \(f_t(G)\) of a graph \(G\) is the least positive integer \(m\) such that however these \(m\) pebbles are placed on the vertices of \(G\), we can move \(t\) pebbles to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we study the generalized Graham’s pebbling conjecture \(f_t(G \times H) \leq f(G)f_t(H)\) for the product of graphs when \(G\) is a complete \(r\)-partite graph and \(H\) has a \(2t\)-pebbling property.

The detour index of a connected graph is defined as the sum of detour distances between all its unordered vertex pairs. We determine the maximum detour index of \(n\)-vertex unicyclic graphs with maximum degree \(\Delta\), and characterize the unique extremal graph, where \(2 \leq \Delta \leq {n-1}\).

In this study, we obtain the relations among \(k\)-Fibonacci, \(k\)-Lucas, and generalized \(k\)-Fibonacci numbers. Then, we define circulant matrices involving \(k\)-Lucas and generalized \(k\)-Fibonacci numbers. Finally, we investigate the upper and lower bounds for the norms of these matrices.

Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a dominating set if every vertex of \(V(G) – S\) is adjacent to some vertices in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality of a dominating set of \(G\). In this paper, we study the domination number of the circulant graphs \(C(n; \{1, 2\})\), \(C(n; \{1, 3\})\), and \(C(n; \{1, 4\})\) and determine their exact values.

The Merrifield-Simmons index of a graph \(G\), denoted by \(i(G)\), is defined to be the total number of its independent sets, including the empty set. Let \(\theta(a_1, a_2, \ldots, a_k)\) denote the graph obtained by connecting two distinct vertices with \(k\) independent paths of lengths \(a_1, a_2, \ldots, a_k\) respectively, we named it as multi-bridge graphs for convenience. Tight upper and lower bounds for the Merrifield-Simmons index of \(\theta(a_1, a_2, \ldots, a_k)\) are established in this paper.

In this paper, it is shown that the graph \(T_{4}(p, q, r)\) is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.

In this paper, using the \(q\)-exponential operator technique to two identities due to Jackson, we obtain some \(q\)-series identities involving \(q\)-analogs of \(_{3}{}{\phi}_{2}\).

We consider words \(\pi_1\pi_2\pi_3\ldots\pi_n\) of length \(n\), where \(\pi_i \in \mathbb{N}\) are independently generated with a geometric probability

\[P({\pi} = k) = p(q)^{k-1} \text{where p + q = 1}. \]

Let \(d\) be a fixed non-negative integer. We say that we have an ascent of size \(d\) or more, an ascent of size less than \(d\), a level, and a descent if \({\pi}_{i+1} \geq {\pi}_i+d \), \({\pi}_{i+1} {\pi}_{i+1} \), respectively.We determine the mean and variance of the number of ascents of size less than \(d\) in a random geometrically distributed word. We also show that the distribution is Gaussian as \(n\) tends to infinity.

The graph \(C_n(d; i, j; P_k)\) denotes a cycle \(C_n\) with path \(P_k\) joining two nonconsecutive vertices \(x_i\) and \(x_j\) of the cycle, where \(d\) is the distance between \(x_i\) and \(x_j\) on \(C_n\). In this paper, we obtain that the graph \(C_n(d; i, j; P_k)\) is strongly \(c\)-harmonious when \(k = 2, 3\) and integer \(n \geq 6\).