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 \(n \geq 6\).

In this paper, we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients.

The concept of integral sum graphs is introduced by Harary \([6]\). A graph \(G\) is an integral sum graph or \(\int\Sigma\)-graph if the vertices of \(G\) can be labelled with distinct integers so that e = uv is an edge of G if and only if the sum of the labels on vertices \(u\) and \(v\) is also a label in G. Xu \([12]\) has shown that the union of any three stars and the union of any number of integral sum trees are integral sum graphs. Xu poses the question as to whether all disconnected forests are integral sum graphs. In this paper, we prove that all banana trees and union of any number of stars are integral sum graphs.

Let \(K_{m} – H\) be the graph obtained from \(K_{m}\) by removing the edges set \(E(H)\) of the graph \(H\) (\(H\) is a subgraph of \(K_{m}\)). We use the symbol \(Z_4\) to denote \(K_4 – P_2\). A sequence \(S\) is potentially \(K_{m} – H\)-graphical if it has a realization containing a \(K_{m} – H\) as a subgraph. Let \(\sigma(K_{m} – H, n)\) denote the smallest degree sum such that every \(n\)-term graphical sequence \(S\) with \(\sigma(S) \geq \sigma(K_{m} – H, n)\) is potentially \(K_{m} – H\)-graphical. In this paper, we determine the values of \(\sigma(K_{r+1} – Z, n)\) for \(n \geq 5r+19, r+1 \geq k \geq 5, j \geq 5\) where \(Z\) is a graph on \(k\) vertices and \(j\) edges which contains a graph \(Z_4\), but not contains a cycle on \(4\) vertices. We also determine the values of \(\sigma(K_{r+1} – Z_4, n)\), \(\sigma(K_{r+1} – (K_4 – e), n)\), \(\sigma(K_{r+1} – K_4, n)\) for \(n \geq 5r+16, r \geq 4\).

A nowhere-zero \(k\)-tension on a graph \(G\) is a mapping from the edges of \(G\) to the set \(\{\pm 1,\pm 2,\ldots,\pm (k-1)\} \subset \mathbb{Z}\) such that, in any fixed orientation of \(G\), for each circuit \(C\) the sum of the labels over the edges of \(C\) oriented in one direction equals the sum of values of the edges of \(C\) oriented oppositely. We show that the existence of an integral tension polynomial that counts nowhere-zero \(k\)-tension on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for tensions on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero tensions on signed graphs with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the tension counting functions, and reciprocity laws that interpret the evaluations of the tension polynomials at negative integers in terms of the combinatorics of the graph.

This paper considered the concepts of monophonic, closed monophonic, and minimal closed monophonic numbers of a connected graph \(G\). It was shown that any positive integers \(m, n, d\), and \(k\) satisfying the conditions that \(4 \leq n \leq m, 3 \leq d \leq k\), and \(k \geq 2m – n + d + 1\) are realizable as the monophonic number, closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Also, any positive integers \(n, m, d\), and \(k\) with \(2 \leq n \leq m, d \geq 3\), and \(k \geq m + d – 1\) are realizable as the closed monophonic number, minimal closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Further, the closed monophonic number of the composition of connected graphs was also determined.

Let \(\Delta(G)\) be the maximum degree of a graph \(G\), and let \(\mathcal{U}(n, \Delta)\) be the set of all unicyclic graphs on \(n\) vertices with fixed maximum degree \(\Delta\). Among all the graphs in \(\mathcal{U}(n, \Delta)\) (\(\Delta \geq \frac{n+3}{2}\)), we characterize the graph with the maximal spectral radius. We also prove that the spectral radius of a unicyclic graph \(G\) on \(n\) (\(n \geq 30\)) vertices strictly increases with its maximum degree when \(\Delta(G) \geq \lceil\frac{7n}{9}\rceil + 1\).