Open research questions in Graph theory and applications

What the literature leaves open

• The paper suggests that future research could focus on experimental verification of the theoretical results. It also mentions the possibility of studying other types of topological indices or nanomaterials.

  Zagreb indices of hyper carbon nanotube graphs · 2026 · DOI

• The paper identifies a gap in the study of topological indices for hyper carbon nanotube graphs. Prior work has focused on other types of nanomaterials or different topological indices.

  Zagreb indices of hyper carbon nanotube graphs · 2026 · DOI

• Further study of the properties of conformally rigid graphs is needed. Investigation of the applications of the results to graph theory and its applications is required.

  Conformal Rigidity and Spectral Embeddings of Graphs · 2026 · DOI

• The existing theory does not explain the conformal rigidity of certain graphs. There is a need to establish new results using the connection between conformal rigidity and symmetrized spectral embeddings.

  Conformal Rigidity and Spectral Embeddings of Graphs · 2026 · DOI

• The problem is NP-hard for certain parameters. The problem has a complex relationship to other graph modification problems. The paper must address an open question in the literature.

  On the parameterized complexity of s-club cluster edge deletion · 2026 · DOI

• Future research can focus on extending the results to more general types of graphs. Future research can focus on applying the results to practical problems. Future research can focus on developing new techniques for analyzing the properties of sign pattern matrices.

  Sign pattern matrices associated with cycle graphs that require algebraic positivity · 2026 · DOI

• The gap is the characterization of sign pattern matrices associated with cycle graphs that require algebraic positivity. The gap is the lack of a technique for analyzing the properties of sign pattern matrices.

  Sign pattern matrices associated with cycle graphs that require algebraic positivity · 2026 · DOI

• Further study of the properties of the clique and hyperedge-based Laplacians. Application of the results to specific complex systems. Extension of the results to more general hypergraph structures.

  Generalizing lattice structures to hypergraphs: spectra of clique and hyperedge-based Laplacians · 2026 · DOI

• The generalization of lattice structures to hypergraphs is less well understood. There is a need for a systematic study of the Laplacian spectra of hyperlattices.

  Generalizing lattice structures to hypergraphs: spectra of clique and hyperedge-based Laplacians · 2026 · DOI

• Future research can focus on studying the properties of the function ϕ. The results can be applied to other areas of graph theory and chemical graph theory.

  Further Results on Bond Incident Degree Indices of Molecular Trees With Perfect Matchings · 2026 · DOI

• The paper identifies a gap in the study of BID indices of molecular trees with perfect matchings. The gap is addressed by establishing unified extremal results for these indices.

  Further Results on Bond Incident Degree Indices of Molecular Trees With Perfect Matchings · 2026 · DOI

• Applying the graph deformation technique to other problems in discrete spectral geometry. Extending the results to other types of graphs or operators.

  Characterization of Green's function of discrete Schrödinger operator on a finite graph by its spanning subgraphs · 2026 · DOI

• The lack of a characterization of the Green's function in terms of geometric properties of the graph. The need for a graph deformation technique to describe the resolvent.

  Characterization of Green's function of discrete Schrödinger operator on a finite graph by its spanning subgraphs · 2026 · DOI

• Future research can focus on extending the method to non-Abelian groups. Future research can focus on applying the method to real-world problems. Future research can focus on developing new methods for graph signal processing.

  Unified Fourier Transform on Graphs Sampled from Stochastic Block Models · 2026 · DOI

• The lack of a method to compute the Fourier transform on graphs sampled from stochastic block models. The need for a foundation for the graphon-driven Fourier transform on graph signals.

  Unified Fourier Transform on Graphs Sampled from Stochastic Block Models · 2026 · DOI

• The paper suggests extending the pointwise analysis to other graph Laplacian methods. The paper suggests applying the method to various applications such as image and speech recognition. The paper suggests investigating the theoretical properties of the kNN graph Laplacian.

  Improved convergence rate of kNN graph Laplacians: differentiable self-tuned affinity · 2026 · DOI

• The paper identifies a gap in the existing literature on graph Laplacians and kNN graphs. The paper addresses the need for a more accurate and efficient method for graph-based data analysis.

  Improved convergence rate of kNN graph Laplacians: differentiable self-tuned affinity · 2026 · DOI

• We resolve an open problem regarding the positive semi-definiteness of $B_α$ and similar matrices.

  Some Spectral Properties of the $B_α$ Matrix of a Graph · 2026

• The gap in prior work is that density does not capture the subtle structure of a network. The gap in prior work is that prior measures of cohesiveness do not capture the concept of k-cores.

  Network structure and minimum degree · 1983 · DOI

• The paper connects k-SI graph bounds to metric properties via transmission and eccentricity but does not address whether similar extremal results hold for k-SI graphs with additional structural constraints such as planarity, bipartiteness beyond complete bipartite graphs, or specific forbidden minors.

  Extremal results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"> <mml:mi>k</mml:mi> </mml:math> -stepwise irregular graphs · 2026 · DOI

• Lemma 5.3 establishes that 2∆(G) - k divides n(G) when gcd(∆(G), k) = 1 and Cd(G) = 2, but the paper provides no analysis of the constraint this divisibility imposes on the possible values of n(G) for given k and ∆(G). The characterization of feasible (n, ∆, k) triples satisfying this divisibility condition is missing.

  Extremal results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"> <mml:mi>k</mml:mi> </mml:math> -stepwise irregular graphs · 2026 · DOI

• We have produced two different constructions of proper binary trees on a given number of leaves which obtain maximum security. Certainly there are other maximal constructions, as indicated in Proposition 12, and it remains an open problem to classify all structures. In addition, counting the number of maximal configurations would result in an answer to the following question. Question 1. Among all proper binary trees of order n, what is the probability that a uniformly chosen tree has maximum security? Our study has largely been restricted to proper binary trees. The history and development of studies on the protection number also began on specific families of trees, only recently handling cases related to general families of trees. This leads to the question below. Question 2. Let X be your favourite family of rooted trees. Classify trees of type X for which security is maximized. Question 3. Considering Proposition 16, can the statement be generalized to other classes of trees, such as rooted k-ary trees, and trees with a given segment sequence. Finally, from a stochastic perspective, we present the following problem. Question 4. Given a tree that obtains maximum security on ℓ ≥ 2 leaves, we label the ℓ leaves with index proportional to the height of the leaf (each leaf is numbered uniquely x0, x1, . . . , xℓ−1 with lower index allocated to lower height). “Grow” each xi into a complete tree of height i. We claim that this is often also a maximal tree. When is this tree maximal, when is it not? As far as we are aware, the security of trees is a new approach to studying the protection number or rank of trees. Therefore, it opens up a number of future directions yet to be explored, and we hope that our results in this direction are just the tip of the research iceberg!

  Characterization of Trees with Maximum Security · 2026 · DOI

• The paper does not provide a complete characterization of all Laplacian integral graphs. The results are limited to the H-join of Laplacian integral graphs.

  On the H-join and H-product of Laplacian and constructably Laplacian integral graphs · 2026 · DOI

• The lack of a characterization of when the H-join of Laplacian integral graphs is Laplacian integral. The need for a necessary and sufficient condition for the H-join of Laplacian integral graphs to be Laplacian integral.

  On the H-join and H-product of Laplacian and constructably Laplacian integral graphs · 2026 · DOI

• The lack of a general result for the sum of the first two largest signless Laplacian eigenvalues. The need for a proof that K+ 1,e(G)-1 is the unique graph with minimum value of f(G) among graphs with e(G) edges.

  Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues · 2026 · DOI

Most-cited papers in Graph theory and applications

• Folding Edges into Vertices: A Machine-Checked Proof of Bass's Determinant Formula for the Ihara Zeta Function in Lean 4 · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 10 citations
• Graphs whose A_α radius does not exceed 2 · Discussiones Mathematicae Graph Theory · 2020 · 10 citations
• Some properties of the eigenvalues of the net Laplacian matrix of a signed graph · Discussiones Mathematicae Graph Theory · 2020 · 6 citations
• More on signed graphs with at most three eigenvalues · Discussiones Mathematicae Graph Theory · 2021 · 6 citations
• Metric dimension and diameter in bipartite graphs · Discussiones Mathematicae Graph Theory · 2020 · 4 citations
• Spectral bounds for the zero forcing number of a graph · Discussiones Mathematicae Graph Theory · 2023 · 3 citations
• Turan's theorem implies Stanley's bound · Discussiones Mathematicae Graph Theory · 2020 · 3 citations
• Correction ^* · Journal of Applied Mathematics Statistics and Informatics · 2022 · 3 citations
• High-ordered spectral characterization of unicyclic graphs · Discussiones Mathematicae Graph Theory · 2023 · 2 citations
• A new method to compute the determinantal polynomial coefficients of a matrix (Symmetric Normalized Kirchhoff matrix) of a complete graph · Journal of Applied Mathematics Statistics and Informatics · 2024 · 2 citations

Most recent work

• Folding Edges into Vertices: A Machine-Checked Proof of Bass's Determinant Formula for the Ihara Zeta Function in Lean 4 · Zenodo (CERN European Organization for Nuclear Research) · 2026
• New upper bounds on Zagreb indices with given domination number · Applied Mathematics and Computation · 2026
• Extremal results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"> <mml:mi>k</mml:mi> </mml:math> -stepwise irregular graphs · Applied Mathematics and Computation · 2026
• Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues · Discrete Applied Mathematics · 2026
• Trees with one as Laplacian eigenvalue with multiplicity two less than the number of pendant vertices · Discrete Mathematics · 2026
• The general energy of a graph · Open Journal of Discrete Applied Mathematics · 2026
• Entropy-Based Topological Characterization of Magnesium Silicide · Punjab University Journal of Mathematics · 2026
• Bounds on F-Index of Tricyclic Graphs with AI Applications · Punjab University Journal of Mathematics · 2026
• Spectral Analysis of Order GCD Graph on Integers Modulo Ring · Pan-American Journal of Mathematics · 2026
• On the Conjugated Chemical Trees of a Given Order and Their Extremum Bond Incident Degree Indices · Contemporary Mathematics · 2026

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