Path-Sum Theorem: Finite Sum Reduction

• The path-sum theorem is a framework that reduces infinite walk-sums in graphs and matrices to finite sums over simple paths with correction terms for cycles.
• It enables efficient computation of matrix functions, such as inverses and exponentials, and facilitates the enumeration of combinatorial structures like Hamiltonian paths and zero-sum subgraphs.
• Its methodology underpins algorithms in graph theory, network science, and statistical physics by transforming complex recurrences into tractable and practical solutions.

The path-sum theorem is a central conceptual framework for expressing combinatorial, algebraic, and analytic quantities associated with paths and walks on graphs, matrices, and discrete structures. Its essential feature is the reduction of summations over families of walks to finite sums over simple paths, augmented by correction terms encoding cycles or external connections. Path-sum identities underpin deep results in algebraic graph theory, combinatorics, and matrix analysis, including zero-sum coloring, matrix function evaluation, simple-path enumeration, and recurrence relations in Bernoulli triangles.

1. Graphical and Algebraic Foundations of Path-Sum Theorems

The path-sum framework begins with the representation of discrete structures — graphs or matrices — in which combinatorial objects such as paths, cycles, and walks carry algebraic weights.

Let $G = (V, E)$ denote a graph, possibly directed or weighted. For the algebraic approach, a "discrete matrix" $M \in \text{Mat}_n(\mathbb{D})$ over a division ring $\mathbb{D}$ can be partitioned as $M_{ij} = E_i M E_j$, generating an associated weighted digraph $G$ where edges represent nonzero matrix blocks. Walks correspond to ordered products within the matrix, and simple paths are walks that do not revisit vertices.

A crucial insight is that summations over all walks — typically infinite due to cycles — can be exactly resummed as finite sums over simple paths, with "dressed" vertex weights that encode the contribution of all cycles or recursive substructures. This foundational fact enables closed-form expressions for various matrix functions (e.g., powers, exponentials, inverses, logarithms) as path-sums over the graph (Giscard et al., 2011).

2. Path-Sum Theorems in Combinatorial Graph Theory

Zero-Sum Path-Sum Theorems

In the context of edge-colored graphs, path-sum formulations yield best-possible conditions for zero-sum subgraphs and zero-sum connectivity.

Given a 2-coloring $f: E(G) \rightarrow \{-1, 1\}$, the zero-sum path-sum framework asserts:

• For complete graphs $K_n$, if $$\min\{e(-1), e(1)\} > \binom{n}{2} - \binom{n-\lfloor\frac{n-3}{4}\rfloor}{2} + c$$, then there exists a spanning path $P$ with $|f(P)| \leq 1$, where $c$ depends on parity (Caro et al., 2020).
• If $\min\{e(-1), e(1)\} \geq \lceil(n+1)/2\rceil$, every pair of vertices is joined by a zero-sum path of length at most 4, and this bound is tight.

These results are established via an interpolation lemma: in any closed family of $m$-edge subgraphs, if two members have weights of opposite sign, then a zero-sum or almost zero-sum member exists. The master theorem extends this to Turán-type conditions, $\mathcal{F}$-decompositions, and covering-family decompositions, with best-possible constants.

Enumeration of Simple Paths

For simple path enumeration in a general (weighted or directed) graph, the path-sum identity expresses the generating function for all simple paths from $u$ to $v$ as

$P_{u,v}(t)\;=\; \sum_{\substack{C\subseteq V\G(C)\text{ weakly connected}\u,v\in C}} \left[(t W_C)^{|C|-1} (I - t W_C)^{|N(C)|}\right]_{u,v},$

where $W_C$ is the restriction of the adjacency matrix to subset $C$, and $|N(C)|$ is the number of outside neighbors (Giscard et al., 2016). The factor $(I-tW_C)^{|N(C)|}$ implements inclusion-exclusion, eliminating walks that are not simple.

This formula generalizes to Hamiltonian paths and cycles, linking their counts to sums over connected dominating sets using signed powers of restricted adjacency matrices. The block-matrix form yields simultaneous enumeration of open and closed Hamiltonian paths.

3. Matrix Function Evaluation via Path-Sum Method

The path-sum theorem provides universal formulas for analytic functions of discrete matrices, even with noncommuting entries. By mapping matrix powers to walks in a weighted digraph and resumming over simple paths, one obtains finite expressions for functions such as $f(M)$:

$$[f(M)]_{ab} = \sum_{P=(a=i_0 \rightarrow \cdots \rightarrow i_\ell=b)} F_G[a]\,M_{i_\ell,i_{\ell-1}}\,F_G[i_{\ell-1}]\cdots M_{i_1,i_0}F_G[b],$$

where $F_G[v]$ is the dressed weight (a continued fraction of cycles based at $v$). Specific cases include:

• Matrix Powers: $(M^q)_{ab}$, with $q \in \mathbb{C}$, given by a path-sum involving an inverse $Z$-transform.
• Matrix Inverses: $(M^{-1})_{ab}$, with signs determined by path length and dressed weights via geometric series resummation.
• Matrix Exponential: $\exp(T M)_{ab}$, computed by Laplace-transforming the path-sum.
• Matrix Logarithm: $(\log M)_{ab}$, as an integral of path-sums using binomial expansion.

These continued-fraction expansions are exact and manifestly finite for finite graphs or matrices, with the inversion height (nesting depth of inverses) tied to graph cycle length. Evaluating minimal inversion-height expressions is NP-complete, as hard as finding longest cycles (Giscard et al., 2011).

A notable application is the explicit algebraic expression for quasideterminants in noncommutative division rings, where $|M|_{ij} = (M^{-1})_{ji}^{-1}$ is readily computed from the path-sum for the inverse.

4. Path-Sum Identities in Bernoulli’s Triangle and Fibonacci Connections

The path-sum theorem extends to combinatorial structures such as Bernoulli’s triangle, yielding closed-form identities for sums over paths that are expressible in terms of Fibonacci numbers. Defining $B^{(2)}_{n,k}$ as the second-order Bernoulli triangle entries,

• The sum along south-west paths from $(n,n)$:

$$S_n = \sum_{k=0}^{\lfloor n/2 \rfloor} B^{(2)}_{n-k,\, n-2k} = 2^{n+1} - F_{n+2},$$

where $F_{m}$ is the $m$th Fibonacci number.

• The sum along south-west paths from $(n,0)$:

$$T_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \sum_{q=0}^k B^{(2)}_{n-k,\,q} = F_{n+3} - 2^{\lfloor (n+1)/2 \rfloor}.$$

These path-sum identities generalize to higher-order Bernoulli triangles, yielding formulas of the form $$T_n^{[m]} = F_{n+2m-1} - 2^{\lfloor(n+1)/2\rfloor} \cdot P_m(\lfloor(n+1)/2\rfloor)$$, where $P_m$ is a polynomial of degree $m-2$ (Neiter et al., 2016). The recurrence structure induced by path-summation mirrors the Fibonacci recurrence, with explicit coefficients derived via auxiliary lemma and inclusion-exclusion.

5. Algorithmic and Applied Consequences

Path-sum identities facilitate algorithmic computation and theoretical analysis across a range of domains:

• Graph Algorithms: The path-sum formulas for simple path enumeration yield efficient algorithms whose complexity depends on the count of connected induced subgraphs, optimizing performance on sparse or bounded-degree graphs (Giscard et al., 2016).
• Network Science: Motif-finding, graph kernels based on all-paths, and accelerated Monte Carlo sampling for self-avoiding walks are grounded in path-sum formalism.
• Statistical Physics and Chemistry: Partition functions for self-avoiding walks on lattices and molecular graphs leverage path-sum enumerations.
• Matrix Analysis: The path-sum approach enables symbolic calculation of matrix functions, especially in noncommutative settings, offering explicit finite continued-fraction representations.
• Combinatorial Number Theory: The link between paths in Bernoulli’s triangles and Fibonacci numbers elucidates the deep interplay between combinatorics and recurrence sequences.

These applications are enabled by the reduction of infinite or intractable walk-sums to finite, structure-preserving path-sums, often expressed in closed form or amenable to recurrence solution.

6. Extensions, Limitations, and Optimality

The optimality of path-sum bounds (e.g., zero-sum path length, coloring thresholds) is established in the literature, with tightness confirmed by explicit constructions (Caro et al., 2020). The machinery generalizes to multigraphs, hypergraphs (with incidence-adjacency matrices), and division ring-valued matrices such as quaternions.

A plausible implication is that recurrence identities derived from path-sum analysis frequently inherit the combinatorial structures of walks and cycles on the underlying graph or matrix partition, and that the inversion-height complexity serves as a critical constraint on symbolic computation in noncommutative algebra.

The path-sum theorem thus serves as a unifying principle in combinatorial analysis, algebraic graph theory, matrix function evaluation, and discrete recurrence, with consequences for both theoretical foundations and practical algorithm design.