Open research questions in Graph theory and applications

What the literature leaves open

• The study of the general energy of graphs is important in mathematical chemistry and network science, but bounds on the general energy were not established. The relationship between the energy, spectral moments, and spectral radius was not well understood.

  The general energy of a graph · 2026 · DOI

• Future research can focus on the application of entropy-based descriptors to other metal silicides. Future research can also explore the relationship between entropy and molecular structure in other materials.

  Entropy-Based Topological Characterization of Magnesium Silicide · 2026 · DOI

• The study of graph irregularity has been a significant research direction, but the concept of k-stepwise irregular graphs has not been fully explored. There is a need to prove the existence of k-SI graphs and provide sharp upper bounds for their maximum degree and size.

  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

• 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?

  Characterization of Trees with Maximum Security · 2026 · DOI

• The gap is that prior work only constructed graphs cospectral with Kp,1q. The gap is that prior work does not determine all graphs cospectral with Kp,kq.

  The Graphs Which Are Cospectral with the Generalized PineappleGraph · 2026 · DOI

• Future research can explore the application of the paper's findings to various fields. It can investigate the relationship between centrality measures and other graph properties. The research can be extended to study more complex graph structures.

  Maximum Non-Self-Centrality Number of Trees With a Fixed Number of Pendant Vertices · 2026 · DOI

• It recognizes the need to study the maximum non-self-centrality number of trees with a fixed number of pendant vertices.

  Maximum Non-Self-Centrality Number of Trees With a Fixed Number of Pendant Vertices · 2026 · DOI

• The paper addresses this gap by deriving formulas for the atom-bond connectivity index, Randic index, and harmonic index.

  Atom-Bond Connectivity Index on a Graph of Finite Free Semilattices · 2026 · DOI

• The limitations are not explicitly stated. The study may be limited to theoretical aspects of graph theory. No specific limitations are mentioned.

  Fractional k-clique metric dimension of (edge) corona products of graphs · 2026 · DOI

• Future research directions are not explicitly stated. The study of fractional k-clique metric dimensions may lead to new research directions. No specific future research directions are mentioned.

  Fractional k-clique metric dimension of (edge) corona products of graphs · 2026 · DOI

• The lack of explicit formula for the Hosoya index of t leaf clover keychain graph. The lack of research on the Hosoya index of t leaf clover keychain graph.

  Hosoya Index of T Leaf Clover Keychain Graph and Its Sequence · 2026 · DOI

• The need for a diagnostic parameter for structural identifiability in molecular and crystalline networks. The lack of a comprehensive understanding of the relationship between the SMD and the topology of crystalline networks.

  Structural identifiability in crystalline molecular networks using strong metric dimension · 2026 · DOI

• The paper does not explicitly state any limitations. However, the research is limited to the study of order GCD graphs of integers modulo ring, and does not consider other types of graphs.

  Structural Properties and Reverse Topological Indices of Order GCD Graphs of Integers Modulo Ring · 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

• Concise closed forms for the effective resistance and spanning trees are not expected in general. The analysis of circulant graphs obtained from complete graphs by deleting edges belonging to a prescribed distance class is limited.

  Effective resistance and spanning trees in complete graphs with distance-class deletions · 2026 · DOI

• The suggested future research directions are not explicitly stated. The study of the p-spectral radius of Berge hypergraphs may lead to further research in graph theory and combinatorics.

  The Extremal p-Spectral Radius of Special Berge Hypergraph · 2026 · DOI

• The paper does not provide a comprehensive study of the applications of the generalized block diagonal Laplacian spectrum. The results of the paper are limited to the study of graphs and may not be applicable to other areas of research.

  Generalized block diagonal Laplacian spectrum of graphs · 2026 · 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

• The method does not address graphs with more than two cycles. The paper does not provide a detailed investigation of the stability of individual cycles. The results are based on the simplex method and may not be generalisable to other methods.

  Complete Heteroclinic Networks Derived from Graphs Consisting of Two Cycles · 2026 · DOI

• To explore the application of the strong metric dimension to other types of crystalline networks. To investigate the relationship between the SMD and other graph-theoretic parameters. To develop a more comprehensive understanding of the structural patterns governing the SMD in crystalline networks.

  Structural identifiability in crystalline molecular networks using strong metric dimension · 2026 · DOI

• Further study of the properties of the Hamming matrix and its applications. Exploration of the implications of the results for various fields, including chemistry, physics, and computer science.

  On the Hamming Spectrum and Hamming Energy of Graphs · 2026 · DOI

• The paper identifies a gap in the understanding of the Hamming spectrum and Hamming energy of graphs. The authors aim to fill this gap by deriving bounds and closed-form expressions for the Hamming spectrum and energy of various graph classes.

  On the Hamming Spectrum and Hamming Energy of Graphs · 2026 · DOI

• To consider joining two complete heteroclinic networks rather than two heteroclinic cycles. To apply the method to more general graphs. To study the stability of individual cycles in heteroclinic networks.

  Complete Heteroclinic Networks Derived from Graphs Consisting of Two Cycles · 2026 · DOI

• The study of finite abelian p-groups and fuzzy subgroups of finite abelian groups. The development of more general subgroup-based and fuzzy graph models.

  Structural Properties and Reverse Topological Indices of Order GCD Graphs of Integers Modulo Ring · 2026 · DOI

• To extend the analysis to more general graphs and distance-class deletions. To apply the results to practical problems in electrical network theory and random walk theory.

  Effective resistance and spanning trees in complete graphs with distance-class deletions · 2026 · DOI

Most-cited papers in Graph theory and applications

• The Centrality Index of a Graph · Psychometrika · 1966 · 1,974 citations
• Theory and practise of the g-index · Scientometrics · 2006 · 1,870 citations
• Network structure and minimum degree · Social Networks · 1983 · 1,470 citations
• Exact and superlative index numbers · Journal of Econometrics · 1976 · 1,169 citations
• Towards a spectral theory of graphs based on the signless Laplacian, I · Publications de l Institut Mathematique · 2009 · 205 citations
• On the Polynomial of a Graph · American Mathematical Monthly · 1963 · 195 citations
• The degree variance: An index of graph heterogeneity · Social Networks · 1981 · 166 citations
• The interaction of size and density with graph-level indices · Social Networks · 1999 · 158 citations
• Linear Dependencies Represented by Chain Graphs · Statistical Science · 1993 · 156 citations
• Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices · Journal of the Serbian Chemical Society · 2012 · 138 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
• A note on the spectral radius and [a,b]-factor of graphs · Discrete Mathematics · 2026
• On Diameter-Constrained Unicyclic Graphs and BID Indices · Mediterranean Journal of Mathematics · 2026
• Solutions to open problems on the exponential augmented Zagreb index · Discrete Mathematics · 2026
• Spectral conditions for graphs to contain k-factors · Discrete Mathematics · 2026
• StereoMolGraph: Stereochemistry-Aware Molecular and Reaction Graphs · Journal of Chemical Information and Modeling · 2026
• Spectral extrema of graphs with fixed size: Forbidden star forests · Discrete Mathematics · 2026
• Input Convex Kolmogorov–Arnold Networks · SIAM Journal on Scientific Computing · 2026
• Spectral radius and rainbow k-factors of graphs · Linear Algebra and its Applications · 2026
• New upper bounds on Zagreb indices with given domination number · Applied Mathematics and Computation · 2026

Related topics in Mathematics