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    Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture
    (Cornell University, 2021-01-19)
    Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.
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    On two conjectures about the intersection of longest paths and cycles
    (Elsevier B.V., 2024-11-01)
    A conjecture attributed to Smith states that every two longest cycles in a k-connected graph intersect in at least k vertices. In this paper, we show that every two longest cycles in a k-connected graph on n vertices intersect in at least min⁡{n,8k−n−16} vertices, which confirms Smith's conjecture when k≥(n+16)/7. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either k≤7 or k≥(n+9)/7.
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    An Unsupervised Model Based on Knowledge Graph and Concepts for Sentiment Analysis
    (Institute of Electrical and Electronics Engineers Inc., 2025)
    Sentiment analysis encompasses various fields such as psychology, marketing, and education, with social media serving as a key platform for gauging public opinion. Recently, graph-based methods have proven to be very useful in representing structured data. This study presents an unsupervised, graph knowledge approach to sentiment analysis that vectorizes nodes representing words and their conceptual connections. Using VADER (Valence Aware Dictionary and sentiment Reasoner) alongside conceptual words such as WordNet and ConceptNet, the method builds a graph of words based on sentiment polarity, capturing both co-occurrence and conceptual relationships. Additionally, a novel Polarity-biased Random Walk algorithm creates polarity-sensitive graph walks, which are vectorized using the Skip-Gram technique. The findings indicate that increasing walk length and the number of node walks, with a bias of 0.95 and employing ConceptNet or WordNet, enhances sentiment classification compared to models like Node2Vec, GraphSAGE, Graph Attention, and Graph Convolutional Networks. Lastly, embeddings generated from the IMDB dataset demonstrate superior accuracy in domain-specific tasks when compared to models such as Word2Vec, FastText, GloVe, and BERT.
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