Chemical indices are introduced to correlate chemical compounds’ physical properties with their structures. Among recently introduced such indices, the eccentric connectivity index of a graph \(G\) is defined as \(\xi^C(G) = \sum_{v \in V(G)} deg(v) ec(v)\), where \(deg(v)\) is the degree of a vertex \(v\) and \( ec(v)\) is its eccentricity. The extremal values of \(\xi^C(G)\) have been studied among graphs with various given parameters. In this note, we study trees with extremal values of the eccentric connectivity index with a given degree sequence. The extremal structures are identified; however, they are not unique.

A \(k\)-L\((d, 1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to \(\{0, 1, \ldots, k\}\) such that \(|f(u) – f(v)| > 1\) if \(d(u,v) = 2\) and \(|f(u) – f(v)| \geq d\) if \(d(u,v) = 1\). The L\((d,1)\)-labeling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has a \(k\)-L\((d, 1)\)-labeling. In this paper, we show that \(2d+2 \leq \lambda(C_m \square C_n) \leq 2d+4\) if either \(m\) or \(n\) is odd. Furthermore, the following cases are determined: (a) \(\lambda_d(C_3 \square C_n)\) and \(\lambda_d(C_4 \square C_n)\) for \(d \geq 3\), (b) \(\lambda_d(C_m \square C_n)\) for some \(m\) and \(n\), (c) \(\lambda_d(C_{2m} \square C_{2n})\) for \(d \geq 5\) when \(m\) and \(n\) are even.

The purpose of this paper is to establish several identities involving \(q\)-harmonic numbers by the \(q\)-Chu-Vandermonde convolution formula and obtain some \(q\)-analogues of several known identities.

It will be proved that the problem of determining whether a set of vertices of a dually chordal graphs is the set of leaves of a tree compatible with it can be solved in polynomial time by establishing a connection with finding clique trees of chordal graphs with minimum number of leaves.

A vertex subset \(F\) is an \(R_k\)-vertex-cut of a connected graph \(G\) if \(G – F\) is disconnected and every vertex in \(G – F\) has at least \(k\) neighbors in \(G – F\). The cardinality of the minimum \(R_k\)-vertex-cut of \(G\) is the \(R_k\)-connectivity of \(G\), denoted by \(\kappa^k(G)\). This parameter measures a kind of conditional fault tolerance of networks. In this paper, we determine \(R_2\)-connectivity and \(R_3\)-connectivity of recursive circulant graphs \(G(2^m, 2)\).

In this paper, we introduce \(h(x)\)-Lucas quaternion polynomials that generalize \(k\)-Lucas quaternion numbers that generalize Lucas quaternion numbers. Also we derive the Binet formula and generating function of \(h(x)\)-Lucas quaternion polynomial sequence.

We determine the crossing numbers (i) of the complete graph \(K_n\) with an edge deleted for \(n \leq 12\) and (ii) of the complete bipartite graph \(K_{m,n}\) with an edge deleted for \(m \in \{3,4\}\) and for all natural numbers \(n$\), and also for the case \(m = n = 5\).

A \(G\)-design is called balanced if the degree of each vertex \(x\) is a constant. A \(G\)-design is called strongly balanced if for every \(i = 1, 2, \ldots, h\), there exists a constant \(C_i\) such that \(d_{A_i}(x) = C_i\) for every vertex \(x\), where \(A_i\) are the orbits of the automorphism group of \(G\) on its vertex-set and \(d_{A_i}(x)\) of a vertex is the number of blocks containing \(x\) as an element of \(A_i\). We say that a \(G\)-design is simply balanced if it is balanced, but not strongly balanced. In this paper, we determine the spectrum for simply balanced and strongly balanced House-systems. Further, we determine the spectrum for House-systems of all admissible indices nesting \(C_4\)-systems.

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In this paper, we determine \(h\)-cacti and \(h\)-cactus chains with the extremal Wiener indices, respectively.

A cyclic coloring is a vertex coloring such that vertices incident with the same face receive different colors. Let \(G\) be a plane graph, and let \(\Delta^*\) be the maximum face degree of \(G\). In 1984, Borodin conjectured that every plane graph admits a cyclic coloring with at most \(\left\lfloor \frac{3\Delta^*}{2} \right\rfloor\) colors. In this note, we improve a result of Borodin et al. [On cyclic colorings and their generalizations, Discrete Mathematics 203 (1999), 23-40] by showing that every plane graph with \(\Delta^* = 6\) can be cyclically colored with 9 colors. This confirms the Cyclic Coloring Conjecture in the case \(\Delta^* = 6\).