相关论文: The zero blocking numbers of generalized Kneser gr…

We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.

The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $(claw, diamond, K_4)$-free graphs. Recently, $3$-colorability has been also considered for $(claw, N_{1,1,1})$-free graphs. Here, a generalised…

In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor…

Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such…

In this paper, we generalize the concept of complete coloring and achromatic number to 2-edge-colored graphs and signed graphs. We give some useful relationships between different possible definitions of such achromatic numbers and prove…

We suggest a new method on coloring generalized Kneser graphs based on hypergraphs with high discrepancy and small number of edges. The main result is providing a proper coloring of K(n, n/2-t, s) in (4 + o(1))(s + t)^2 colors, which is…

We show that the full-flag Johnson graph has spectral gap equal to that of its Schreier quotient arising from the point-stabiliser equitable partition. Our results confirm two conjectures posed by Huang, Huang, and Cioab\u{a}, which imply…

We define a Johnson graph code as a subspace of labelings of the vertices in a Johnson graph with the property that labelings are uniquely determined by their restriction to vertex neighborhoods specified by the parameters of the code. We…

Graph neural networks (GNNs) are the most widely adopted model in graph-structured data oriented learning and representation. Despite their extraordinary success in real-world applications, understanding their working mechanism by theory is…

The Johnson graph $J(n,i)$ is defined to the graph whose vertex set is the set of all $i$-element subsets of $\{1,\ldots,n\}$, and two vertices are joined whenever the cardinality of their intersection is equal to $i-1$. In Ramras and…

We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify…

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin holds for all but finitely many graphs. We also use our method to prove…

For graph classes $P_1,...,P_k$, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph $G$ can be partitioned into subsets $V_1,...,V_k$ so that $V_j$ induces a graph in the class $P_j$…

The so called generalized down-up algebras are revisited from a viewpoint of Gr\"obner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $\lambda\omega\ne 0$), and…

We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the…

There is a profound connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and solving various challenging graph-related problems. Conversely, graph theory provides a rich set of…

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We also…

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that…

In this paper, we determine the structures of zero-divisor semigroups whose graph is $K_n + 1$, the complete graph $K_n$ together with an end vertex. We also present a formula to calculate the number of non-isomorphic zero-divisor…

We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.