Rank 3 Self-Loop Graphs in Algebraic Spectral Theory

• Rank 3 self-loop graphs are defined as graphs whose modified adjacency matrix, after adding self-loops, exhibits exactly three distinct eigenvalues, offering a clear spectral signature.
• They bridge symmetric graph families and combinatorial designs by connecting strongly regular graphs, divisible design graphs, and applications from quantum gravity to molecular energy descriptors.
• Recent classifications and constructions via cyclic and join operations provide actionable insights into spectrum, energy metrics, and topological invariants.

Rank 3 self-loop graphs are graphs whose adjacency or Laplacian matrices exhibit rank 3, or more generally, graphs possessing only three distinct eigenvalues, after accounting for possible self-loops. Such structures represent a tight interface between highly symmetric graph families (like strongly regular graphs and divisibly designed graphs) and explicit spectral and combinatorial constraints, making them central objects in algebraic and spectral graph theory, permutation group theory, and mathematical physics.

1. Algebraic and Combinatorial Definition

A rank 3 self-loop graph is a graph $G_S$ formed from a simple graph $G$ by attaching a self-loop at every vertex in some subset $S\subseteq V(G)$. The adjacency matrix of $G_S$ is given by: $A(G_S) = A(G) + I_S$ where $I_S$ is a diagonal matrix with $1$ in positions corresponding to $S$, and zeros elsewhere. The graph rank is taken to be the rank of $A(G_S)$, or equivalently the dimension of its column space.

Such a graph is called a "rank 3 graph" when $A(G_S)$ has exactly three distinct eigenvalues, extending the classical notion for simple graphs and strongly regular graphs. In the context of permutation groups, rank 3 graphs correspond to orbital graphs of groups $G$ having exactly three orbits on ordered pairs, ensuring their automorphism group acts transitively on edges and non-edges.

2. Classification and Structural Results

Several papers provide precise classification and construction results for rank 3 self-loop graphs:

• Triangle-Free Cyclic Graphs: For cycles $C_n$, it is shown that attaching self-loops to any subset $S$ yields a rank at least $4$ for $n\geq 5$; only for 4-cycles $C_4$ do rank 3 self-loop graphs occur. Families of triangle-free, connected, rank 3 cyclic graphs are constructed using graph join operations from $(C_4)_S$ with $|S|=1$ or $|S|=2$ (Lim, 24 Sep 2025). These are the only such triangle-free cyclic self-loop graphs of rank 3.
• Strongly Regular and Divisible Design Graphs with Self-Loops (LDDG's): In the theory of LDDGs, those with precisely three distinct eigenvalues are isomorphic to complete multipartite graphs $K_{n,\dots,n}$ with respect to their canonical partition. Many examples derive from symplectic, orthogonal, and unitary polarities in projective geometries, which sometimes yield self-loops at every vertex. When regularity, divisibility, and spectral conditions align, such LDDGs become rank 3 graphs (Bhowmik et al., 6 May 2025).
• Ideal Whitehead Graphs and Self-Loop Roses in $\mathrm{Out}(F_3)$: The so-called rank 3 "rose" (one vertex with three loops) provides a combinatorial template for data encoded in train track representatives of fully irreducible automorphisms in $\mathrm{Out}(F_3)$ (Pfaff, 2013). There are precisely 21 connected, simplicial 5-vertex graphs possible; exactly 18 occur as the ideal Whitehead graph for some fully irreducible $\phi$, establishing fine invariants for outer automorphisms.

3. Spectral Properties and Energy

Self-loops alter the spectral properties of graphs in carefully controlled ways:

• Spectrum of Laplacians: For a graph $G$ (possibly with self-loops), the Laplacian is $L(G) = L(G^\circ) + \sum_{(i,i)\in E} e_i e_i^T$, where $G^\circ$ is the graph with self-loops removed. The paper (Acikmese, 2015) introduces pseudo-connected graphs, showing that the Laplacian is positive definite whenever every connected subgraph has at least one self-loop: $v^T L(G) v > 0$ for any nonzero $v$. The spectrum satisfies

$$\sigma(L(G)) \subseteq \sigma(L(\hat G)) \cap [0, 2d_{\max}(G^\circ) + 1]$$

where $\hat G$ is a lifted graph without self-loops.

• Graph Energy: The energy $\mathcal E(G_S)$ of a self-loop graph, defined as the sum of the absolute values of the eigenvalues of $A(G_S)$, is explicitly sensitive to the inclusion of self-loops (Rakshith et al., 23 May 2024). For any nonempty $S$, there exists $S$ so that $\mathcal E(G_S) > \mathcal E(G)$, confirming a conjecture of Akbari et al. Explicit constructions yield equienergetic but non-isomorphic self-loop graphs.
• Twisted Moments and Closed Walks: Exact formulae for closed $k$-walks (especially $k=3,4$) on $G_S$ are available in terms of vertex degrees and graph structure. The paper (Lim, 21 Sep 2025) develops twisted moment quantities: $$\mathcal M_q(G_S) = \sum_{i=1}^n |\lambda_i(G_S) - \frac{\sigma}{n}|^q$$ with proven ratio inequalities: $$\frac{\mathcal M_1}{\mathcal M_0} \leq \frac{\mathcal M_2}{\mathcal M_1} \leq \frac{\mathcal M_3}{\mathcal M_2} \leq \cdots$$ allowing lower bounds on energy and direct spectral constraints, which are especially relevant for rank 3 cases where energy and spectral moments are tightly related.

4. Polynomial and Topological Invariants

Extensions of classical graph invariants to the higher-rank, self-loop setting have strong topological and combinatorial consequences:

• Generalized Bollobás-Riordan Polynomials: The BR polynomial is extended to rank 3 weakly-colored stranded graphs—graphical models with intricate strand and bubble structure appearing in tensor theories of quantum gravity (Avohou et al., 2013) and (Avohou, 2015). The invariant $\mathcal T_G(x, y, z, s, w, q, t)$ sums over spanning "cut" subgraphs, weighting by quantities such as the number of vertices, edges, closed faces, bubbles, boundary components, and half-edges. Contraction/cut recursion relations generalize the deletion/contraction rules of the Tutte and original BR polynomials.
• Connections to Topological Quantum Invariants: Invariants arising from graph complexes of decorated trivalent graphs (allowing self-loops) underpin perturbative Chern-Simons theory (Kodani et al., 2023). An adapted propagator condition eliminates dumbbell (self-loop) contributions in topological invariants for framed 3-manifolds, leading to generating series formulated solely from graphs without self-loops, while maintaining equivalence with self-loop-including complexes.

5. Representative Examples and Constructions

• Graph Joins for Cyclic Graphs: All triangle-free connected rank 3 cyclic self-loop graphs (for order at least 4) are constructed from $(C_4)_S$ using join operations over independent sets, leading to precisely six non-isomorphic families (Lim, 24 Sep 2025). For $n\geq 5$ cycles, no such configuration attains rank 3.
• Divisible Design Graphs from Geometries: Symplectic polarities in projective space $PG(2n-1,q)$ yield LDDGs with a loop at every vertex; spectral analysis shows some collapse to rank 3 (Bhowmik et al., 6 May 2025).
• Equiangular Tight Frames (ETFs): Primitive rank 3 graphs produce real ETFs when embedded as spherical configurations—their combinatorial and spectral data guarantees equiangularity and tightness, with switching operations on descendants (by vertex removal) frequently regenerating rank 3 graphs (Bannai et al., 2022).

6. Applications and Connections

Rank 3 self-loop graphs are pivotal in:

• The spectral paper of molecular and chemical graphs: energy descriptors based on adjacency spectrum reflect physical and chemical properties, with self-loops modeling heteroatomic effects (Rakshith et al., 23 May 2024).
• The analysis of synchronization and separation in permutation group theory: such graphs are the archetype for sharply distinguished synchronizing primitive groups, including affine types that are not QI, with concrete examples classified (Bamberg et al., 2022).
• The combinatorial topology of quantum gravity: higher-dimensional polynomial invariants (generalizations of Tutte and Bollobás-Riordan) encode essential features of quantum geometric models (Avohou et al., 2013, Avohou, 2015).
• Frame theory and optimal spherical codes: primitive rank 3 graphs achieve ETFs attaining fundamental bounds of equiangularity and frame tightness (Bannai et al., 2022).

7. Open Problems and Future Directions

Key open directions and questions include:

• Complete classification of all rank 3 self-loop graphs, particularly extending beyond triangle-free cycles and considering more complex base graphs and self-loop configurations (Lim, 24 Sep 2025).
• Structural constraints and parameter spaces in divisible design graphs, including the interplay between self-loops and eigenvalue multiplicities (Bhowmik et al., 6 May 2025).
• Extension of spectral bounds and twisted moment inequalities to weighted/self-loop graphs for broader classes, with implications in quantum information and spectral geometry (Lim, 21 Sep 2025).
• Connections between higher-rank polynomial invariants and physical topological invariants for quantum field models (Avohou et al., 2013, Avohou, 2015, Kodani et al., 2023).
• Explicit constructions of equienergetic and synchronizing rank 3 self-loop graphs for combinatorial and coding purposes (Rakshith et al., 23 May 2024, Bamberg et al., 2022).

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