arXiv subjects

Annayat Ali

Publications and source records attributed to Annayat Ali.

3 recordsLinked to original sources

arXiv2026

$\lambda$-Chromatic Polynomials and Polytope Geometry

In this paper, we investigate the notion of the \textit{$\lambda$-chromatic polynomial} of a graph, which enumerates the number of distinct $L(2,1)$-colorings using colors from a prescribed finite set. We prove that the $\lambda$-chromatic polynomial of a graph with $n$ vertices is a monic polynomial of degree $n$ and provide a combinatorial interpretation via lattice point enumeration within the framework of inside-out polytopes. Moreover, we compute the $\lambda$-chromatic polynomial of complete graphs $K_n$ using lattice path enumeration, and we develop a block-gap technique to derive the $\lambda$-chromatic polynomials for complete bipartite and multipartite graphs. Our approach unifies geometric, combinatorial, and algebraic methods to provide a systematic treatment of $\lambda$-colorings across various families of graphs.

math.CO↗

arXiv2023

L(2, 1)-labeling of some zero-divisor graphs associated with commutative rings

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a simple graph, an $L(2,1)$-labeling of $\mathcal{G}$ is an assignment of labels from nonnegative integers to vertices of $\mathcal{G}$ such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The $λ$-number of $\mathcal{G}$, denoted by $λ(\mathcal{G})$, is the smallest positive integer $\ell$ such that $\mathcal{G}$ has an $L(2,1)$-labeling with all labels as members of the set $\{0,1, \dots,\ell\}$. The zero-divisor graph of a finite commutative ring $R$ with unity, denoted by $Γ(R)$, is the simple graph whose vertices are all zero divisors of $R$ in which two vertices $u$ and $v$ are adjacent if and only if $uv = 0$ in $R$. In this paper, we investigate $L(2,1)$-labeling of some zero-divisor graphs. We study the partite truncation, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between $λ$-numbers of the graph and its partite truncated one. We make use of the operation partite truncation to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring.

math.CO↗

arXiv2023

On the expansion constant and distance constrained colourings of hypergraphs

For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that at least differ by k. The smallest positive integer δ such that G permits an L(h, k)-colouring with maximum colour δ is known as the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by λ_{h,k}(G). In this paper, we discuss some interesting invariants in hypergraphs. In fact, we study the relation between the spectral gap and L(2, 1)-chromatic number of hypergraphs. We derive some inequalities which relates L(2, 1)-chromatic number of a k-regular simple graph to its spectral gap and expansion constant. The upper bound of L(h, k)-chromatic number in terms of various hypergraph invariants such as strong chromatic number, strong independent number and maximum degree is obtained. We determine the sharp upper bound for L(2, 1)-chromatic number of hypertrees in terms of its maximum degree. Finally, we conclude this paper with a discussion on L(2, 1)-colouring in cartesian product of some classes of hypergraphs.

math.CO↗