The singular difference graph, denoted by $\Gamma$, of the vector space of square matrices over a field is a graph whose vertex set is the set of all elements of the vector space, where two distinct vertices are adjacent if and only if the difference of the corresponding matrices is singular. In this paper, we investigate fundamental graph-theoretic properties of $\Gamma$, including connectivity, diameter, regularity, the Eulerian property, independence number, clique number, and domination number. We show that $\Gamma$ is a connected regular graph with diameter two. Over finite fields, we obtain an explicit formula for the degree of each vertex and characterize precisely when $\Gamma$ is Eulerian. We determine the independence number and clique number and provide explicit constructions attaining these values using companion matrices of irreducible polynomials. We also construct an explicit dominating set, yielding an upper bound for the domination number.
For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normal...
A graphic arrangement $\A_G$ associated with a simple graph $G$ is a classical and well-studied object in the theory of hyperplane arrangements. In this note, we show that, for a connected graph $G$, a slight modification of the logarithmic vector field $D(\A_G)$ of $\A_G$ is isomorphic to the face ring of a certain si...
Let N =Yti=1pnii , t ≥ 2,
where the primes p1, . . . , pt are distinct and ni ≥ 1, and let R = Z/NZ. The nonzero zero-divisors of R are partitioned by their truncated prime-adic valuation vectors. This paper develops the resulting valuation-layer description of the zero-divisor graph Γ(R). A complete formula is obtain...
Presley Kiplagat, Lao Hussein Mude, Zachary Kayiita· Asian Research Journal of Ma...· 0 citations
The zero-divisor graph of a commutative ring provides a natural connection between algebraic and graph-theoretic structures. Although extensive research has been conducted on the algebraic and combinatorial properties of zero-divisor graphs, their connectivity and metric properties over finite semilocal rings remain co...
Presley Kiplagat· Earthline Journal of Mathema...· 0 citations
The generating graph $\Gamma(G)$ of a group $G$ is the graph whose vertex set is $G$, where two distinct vertices are adjacent if and only if they generate $G$. In this paper, we systematically study the structure of generating graphs of finite abelian groups (non-cyclic) and determine the set of all generating pairs....
The zero divisor graph $\Gamma(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $\Gamma(R)$ for two classes of finite commutative rings: reduced rings and quo...
Marco Caoduro, Meike Neuwohner· 0 citations
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