Geometric multiplicity of unitary non-backtracking eigenvalues

TL;DR

This work characterizes unit-modulus eigenvalues of the non-backtracking matrix $\mathbf{B}$ for undirected graphs, proving they must be roots of unity and that their eigenfunctions live on subgraphs that are graph subdivisions. It develops a non-matrix framework based on NB chains, subdivision theory, and gluing to determine the geometric multiplicities of these eigenvalues with closed formulas, including a linear-time algorithm. The key contributions include precise AM/GM results for the real case, a structural description of supports for nonreal unitary eigenvalues, a subdivision-based spectral relation, and an exact, computation-friendly formula for $\mathrm{GM}_X(\lambda)$ in terms of subdivision components $S_q$. The results enable efficient, structure-aware spectral analysis of NB eigenvalues and have implications for graph zeta functions and non-backtracking-based graph algorithms.

Abstract

We completely characterize the conditions under which a complex unitary number is an eigenvalue of the non-backtracking matrix of an undirected graph. Further, we provide a closed formula to compute its geometric multiplicity and describe an algorithm to compute this multiplicity without making a single matrix computation. The algorithm has time complexity that is linear in the size of the graph.

Figures (9)

• Figure 2.1: Action of non-backtracking matrix.a) Edge color represents the value assigned to that directed edge by the function $\mathbf{v}$. b)$\mathbf{B v}$ aggregates the values along all incoming edges, except for the backtrack $\mathbf{v}_{l \to k}$. Overlapping arrows represent the sum of the corresponding color-coded quantities. c)$\mathbf{B}^{2} \mathbf{v}$ aggregates the values along all NB walks of length $3$.
• Figure 4.1: The action of $\mathbf{B,B^{*},B^{*}B}$ and $\mathbf{BB^{*}}$ on a function $\mathbf{v}$. a) The value of $\mathbf{v}$ at each directed edge is color-coded. b) The value of $\left( \mathbf{B v}\right)_{k \to l}$ equals the sum of the values along edges incoming to $k$, ignoring the backtrack; cf. \ref{['fig:nbm-doodle']}. Overlapping arrows represent the sum of the corresponding color-coded quantities. c-e) The result of applying $\mathbf{B^*, B^* B, B B^*}$ is similarly depicted.
• Figure 4.2: Example of leakiness. The nodes in $\mathcal{D}$ induce a cycle. The node $k\in\mathcal{D}$ has a neighbor $l\notin\mathcal{D}$. Left: a function $\mathbf{v}$ supported on $\mathcal{D}$. Right: if $\mathbf{v}$ is an eigenfunction, the sum of all values incoming to $k$ must be zero. If $\left( \mathbf{Bv} \right)_{k \to l}$ is non-zero, we say that $\mathbf{v}$leaks out of $\mathcal{D}$ via $k$. Nodes with degree $2$ can never leak.
• Figure 5.1: NB chains and graph subdivisions. A graph $G$ (top) and its $p$-subdivision (bottom) for the case $p=2$. The length of every chain in the subdivision is a multiple of $p$. a) If the graph has no nodes of degree $2$, then every chain in the subdivision has length $p$. b) Since $G$ has nodes of degree $2$, its subdivision has chains of length $2p$. c) In general, the maximum length of a NB chain in the subdivision is $p$ times the maximum length of a NB chain in the original graph $G$.
• Figure 6.1: Non-leaky gluing and detaching. For each row, gluing $G$ to $H$ by identifying the nodes by color, except for the gray nodes, produces the graph $X$. The colored nodes in the left column comprise (a subset of) the set $N \subset V(G)$. The three rows produce the same graph $X$ via different gluing operations. Conversely, when analyzing $X$, one can detach a subgraph $G$ to study the (non-leaky) eigenfunctions supported on it. a) Gluing an odd-length circle. The gluing equivalence relation in this case has one equivalence class comprised of the two green nodes, one in $G$ and one in $H$. b) Gluing an even-length circle. The gluing has two non-trivial equivalence classes: one containing the two purple nodes and another containing the two green nodes. c) Gluing a graph with two cycles. The gluing has two non-trivial equivalence classes: one containing the two pink nodes and another containing the two green nodes. a-c) In each row, the gluing is such that each gray node belongs to a trivial equivalence class, i.e. a class that contains a single element.

Theorems & Definitions (80)

• Remark 1: Circle graphs
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• proof
• Proposition 3.5