The main aim of this article is to examine the inter-relationships among the top cryptocurrencies on the crypto stock market in the presence and absence of the COVID-19 pandemic. The nine chosen cryptocurrencies are Bitcoin, Ethereum, Ripple, Litecoin, Eos, BitcoinCash, Binance, Stellar, and Tron and their daily closing price data are captured from coinmarketcap over the period from 13 September 2017 to 21 September 2020. All of the cryptocurrencies are integrated of order 1 i.e., I(1). There is strong evidence of a long-run relationship between Bitcoin and altcoins irrespective of whether it is pre-pandemic or pandemic period. It has also been found that these cryptocurrencies’ prices and their inter-relationship are resilient to the pandemic. It is recommended that when the investors create investment plans and strategies they may highly consider Bitcoin and altcoins jointly as they give sustainability and resilience in the long run against the geopolitical risks and even in the tough time of the COVID-19 pandemic.

This study examines higher education through data-mining methodologies, aiming to uncover key themes and sentiments in global discourse. Utilizing sentiment analysis and topic modeling, the research analyzes 157,943 tweets from 84,423 unique users over a five-month period (January to May 2023). This period was selected, coinciding with the rise of artificial intelligence (AI) tools, particularly ChatGPT. The study investigates the discussions, emotional tones, and dominant topics shaping the global narrative of higher education within X (Twitter) data. Key findings include the geographical distribution of tweets and the most frequent positive and negative perceptions. It also addresses critical issues such as affordability, accessibility, and funding in higher education. Furthermore, the data shows public reactions to AI in higher education are initially negative, while higher education tweets are primarily characterized by positivity and optimism. The higher education tweets are mainly posted on the weekend, with decreased activity during weekdays. This research provides insights into the evolving higher education landscape amid rapid technological advancements.

The distance $d(v_a,v_b)$ between two vertices of a simple connected graph G is the length of the shortest path between $v_a$ and $v_b$. Vertices $v_a,v_b$ of G are considered to be resolved by a vertex v if $d(v_a,v)\ne d(v_b,v)$. An ordered set $W=\{v_1,v_2,v_3,\dots ,v_s\}\subseteq V\left(G\right)$ is said to be a resolving set for G, if for any $v_a,v_b\in V\left(G\right),$∃$v_i\in W\ni d(v_a,v_i)\ne d(v_b,v_i)$. The representation of vertex v with respect to W is denoted by $r\left(v\right|W)$ and is an s-vector(s-tuple) $(d(v,v_1),d(v,v_2),d(v,v_3),\dots ,d(v,v_s))$. Using representation $r\left(v\right|W)$, we can say that W is a resolving set if, for any two vertices $v_a,v_b\in V\left(G\right)$, we have $r\left(v_a\right|W)\ne r\left(v_b\right|W)$. A minimal resolving set is termed a metric basis for G. The cardinality of the metric basis set is called the metric dimension of G, represented by $dim\left(G\right)$. In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.

The distance between two vertices of a simple connected graph G, denoted as $d(u,v)$, is the length of the shortest path from u to v and is always symmetrical. An ordered subset $W=\left\{w_1,w_2,w_3,\cdots ,w_k\right\}$ of $V\left(G\right)$ is a resolving set for G, if for ∀$u,v\in V\left(G\right)$, there exists $w_i\in W$ ∋ $d(u,w_i)\ne d(v,w_i)$. A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as $dim\left(G\right)$. For the graph $G_1=(V_1,E_{1,})$ and $G_2=(V_2,E_2)$, their join is denoted by $G_1+G_2$. The vertex set of $G_1+G_2$ is $V_1\cup V_2$ and the edge set is $E=E_1\cup E_2\cup \left\{uv,u\in V_1,v\in V_2\right\}$. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.

The main aim of this article is to examine the inter-relationships among the top cryptocurrencies on the crypto stock market in the presence and absence of the COVID-19 pandemic. The nine chosen cryptocurrencies are Bitcoin, Ethereum, Ripple, Litecoin, Eos, BitcoinCash, Binance, Stellar, and Tron and their daily closing price data are captured from coinmarketcap over the period from 9/13/2017 to 9/21/2020. To determine the order of integration, the Augmented Dickey-Fuller (ADF), Phillips-Perron (PP), and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) unit root tests were carried out, and all of the cryptocurrencies were found to be integrated of order 1 i.e. I(1). It has been found that there is strong evidence of a long-run relationship between Bitcoin and altcoins irrespective of whether it is pre-pandemic or pandemic period. Hence these cryptocurrencies' prices and their inter-relationship are resilient to the pandemic.