On Spectral Graph Determination

TL;DR

The paper surveys spectral graph determination, detailing when graphs are uniquely identified by one or more matrix spectra ($A$, $L$, $Q$, $\oldsymbol{\mathcal{L}}$) and when cospectral nonisomorphic graphs (NICS) arise. It synthesizes classical results and modern developments, including proofs that Turán graphs are determined by their adjacency spectrum and that several graph families (stars, complete bipartite, line graphs, friendship graphs, strongly regular graphs) have nuanced DS/NICS status across different matrices. It also presents a toolbox of graph-operations (coalescence, Seidel switching, Godsil–McKay method, duplications, subdivisions, and various joins) to construct cospectral graphs, with explicit constructions of irregular X-NICS graphs and discussions of open questions. The work highlights Haemers' conjecture—that almost all graphs are adjacency-DS—and summarizes progress establishing both DS and NICS phenomena across a wide range of structured graphs and matrix representations, underscoring the interplay between symmetry, spectrum, and graph structure. The findings have implications for chemistry, physics, and network analysis by clarifying the limits of spectral characterization and guiding future theoretical and constructive directions.

Abstract

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph's spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights.

Figures (17)

• Figure 1: The graphs $\mathop{\mathrm{\mathsf{S}}}\nolimits_{4} = \mathop{\mathrm{\mathsf{K}}}\nolimits_{1,4}$ and $\mathop{\mathrm{\mathsf{C}}}\nolimits_{4} \space \dot{\cup} \space \mathop{\mathrm{\mathsf{K}}}\nolimits_{1}$ (i.e., a union of a 4-length cycle and an isolated vertex) are cospectral and nonisomorphic graphs ($\mathop{\mathrm{\mathbf{A}}}\nolimits$-NICS graphs) on five vertices. These two graphs therefore cannot be determined by their adjacency matrix.
• Figure 2: $\{\mathop{\mathrm{\mathbf{A}}}\nolimits, \mathop{\mathrm{\mathbf{L}}}\nolimits, \mathop{\mathrm{\mathbf{Q}}}\nolimits, \bf{\mathcal{L}}\}$-NICS regular graphs with $10$ vertices. These cospectral graphs are nonisomorphic because each of the two blue edges in $\mathsf{G}$ belongs to three triangles, whereas no such an edge exists in $\mathsf{H}$.
• Figure 3: The friendship (windmill) graph $\mathop{\mathrm{\mathsf{F}}}\nolimits_{4}$ has 9 vertices, 12 edges, and 4 triangles.
• Figure 4: The duplication graph $\mathrm{Du}(\mathop{\mathrm{\mathsf{C}}}\nolimits_{5})$ (see Definition \ref{['definition: duplication graph']}).
• Figure 5: The corona graph ${\mathop{\mathrm{\mathsf{C}}}\nolimits_{4}} \circ {(2\mathop{\mathrm{\mathsf{K}}}\nolimits_{1})}$ (see Definition \ref{['definition: corona graph']}) consists of a single copy of $\mathop{\mathrm{\mathsf{C}}}\nolimits_{4}$ (represented by the black vertices) and four copies of $2\mathop{\mathrm{\mathsf{K}}}\nolimits_{1}$ (represented by the red vertices).

Theorems & Definitions (158)

• Definition 2.1
• Theorem 2.2: Theorem on the Schur complement Schur1917
• Theorem 2.3: Cauchy Interlacing Theorem ParlettB1998
• Definition 2.4: Completely Positive Matrices
• Definition 2.5: Positive Semidefinite Matrices
• Proposition 2.6
• Corollary 2.7
• Remark 2.8
• Definition 2.9: Complement of a graph
• Definition 2.10: Disjoint union of graphs