arXiv subjects

Vaibhav Suvagiya

Publications and source records attributed to Vaibhav Suvagiya.

3 recordsLinked to original sources

arXiv2026

Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models

We develop an affine $\mathbb F_2$ framework for structured signings of regular graphs. A family-averaging identity converts even spectral moments into parity-weighted closed-walk counts supported on the span of prescribed short even cycles, while a kernel-averaged Ihara identity gives the corresponding decomposition at the non-backtracking level. We give a finite-scale bounded-rank counting estimate and a conditioning corollary showing that, on bicycle-free graph sequences, any parity family of uniformly bounded codimension contains near-Ramanujan signings whenever the corresponding random-signing theorem applies. The latter is a transfer statement rather than a new concentration theorem. Finally, on $C_n(1,2)$ for even $n\ge10$, the quadrilateral-unbalanced family has exactly four switching classes and its twisted classes attain $ρ_-(n)=2\sqrt{\cos^2(π/n)+\cos^2(2π/n)}$; a period-$8$ signing has spectral radius $r_*=2.793604493334841\ldots$ for every positive multiple of $8$. Thus for $n=8m\ge32$ the constrained minimum is strictly larger than a value attained by an unrestricted signing, while equality of $r_*$ with the unrestricted minimum remains conjectural.

math.CO↗

arXiv2026

Signed circulants at the Ramanujan bound

For the circulant graph $C_n(1,2)$ with $n\ge10$ even, the $\F_2$ system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is $+1$ on step-$1$ edges and $(-1)^i$ on step-$2$ edges has spectrum $\{\pm2\sqrt{\cos^2θ_k+\cos^2 2θ_k}\}$ and spectral radius exactly $2\sqrt2$, well below the Kesten bound $2\sqrt3$; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by $(τ_0,α)$ and the spectral radius depends only on the Hamilton-cycle holonomy $α$; and that the two twisted classes attain $ρ_-(n)=2\sqrt{\cos^2(π/n)+\cos^2(2π/n)}<2\sqrt2$. Exhaustive enumeration of all $2^{n+1}$ switching classes for $n\in\{8,10,12,14,16,18\}$ shows that $ρ_-(n)$ is the global minimum in every case, and we conjecture this for all even $n$; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd $n$ the quadrilateral system is inconsistent.

math.CO↗

arXiv2026

Two-distance and list-two-distance coloring of cacti: the subcubic case and the C5 obstruction

The square $G^2$ of a graph joins two vertices at distance at most two; a proper coloring of $G^2$ is a 2-distance coloring of $G$. For a cactus $G$ (every edge on at most one cycle) we determine both the 2-distance chromatic number $χ(G^2)$ and the choice number $\mathrm{ch}(G^2)$ exactly: they are always equal, and the common value is $Δ+1$ if $Δ\ge4$, is 4 if $Δ=3$ and $G$ has no block equal to $C_5$, is 5 if $Δ=3$ and $G$ has a $C_5$ block, and is the classical value if $Δ\le2$. For $Δ\ge6$ the value $Δ+1$ is already known, since cacti are outerplanar and hence $K_{2,3}$-minor-free (Hetherington-Woodall; Agnarsson-Halldorsson). Our contribution is the small-degree regime. For subcubic cacti we obtain a complete classification in both the ordinary and list settings, with the 5-cycle as the unique obstruction; the list statement has no prior analogue and is a genuine positive instance of the List Square Coloring Conjecture, which is false in general. A single elimination order then handles all $Δ\ge4$ uniformly and, in particular, settles the two cases $Δ\in\{4,5\}$ that the superclass bounds leave at $Δ+2$. The number 5 turns out to be one obstruction wearing three disguises: $C_5^2=K_5$, the Frobenius number of $\{3,4\}$, and a degenerate $K_4$ list-coloring instance.

math.CO↗