We prove the Kauffman–Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

Let G = (V, E) be a graph with v vertices and e edges. An (a, d)-vertex-antimagic total labeling is a bijection λ from V (G) ∪ E(G) to the set of consecutive integers 1, 2, . . . , v + e, such that the weights of the vertices form an arithmetic progression with the initial term a and common difference d. If λ(V (G)) = {1, 2, . . . , v} then we call the labeling a super (a, d)-vertex-antimagic t...

We generalize the Harary-Sachs theorem to k-uniform hypergraphs: codegree-d coefficient of characteristic polynomial a uniform hypergraph H can be expressed as weighted sum subgraph counts over certain multi-hypergraphs with d edges. include detailed description aforementioned and formula for their corresponding weights.

Topological indices are the numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Graph theory is a delightful playground for the exploration of proof techniques in Discrete Mathematics and its results have applications in many areas of sciences. One of the useful indic...

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric r-dissections of polygons and planar S-dissections are obtained.

In this paper we will investigate the edge-chromatic automorphism groups for two classes of graphs. Chromatic automorphism groups have been studied by a number of people, including Chvatal [4] and Baumann et al. [l, 21. Characterizations will be given for these groups. Note that all arithmetic will be done modp, p not necessarily prime. The particular p will be clear from the context. These gra...

Recently, Hua et al. defined a new topological index based on degrees and inverse of distances between all pairs of vertices. They named this new graph invariant as reciprocal degree distance as 1 { , } ( ) ( ( ) ( ))[ ( , )] RDD(G) = u v V G d u  d v d u v , where the d(u,v) denotes the distance between vertices u and v. In this paper, we compute this topological index for Grassmann graphs.

Two graphs G and H are hypomorphic if there exists a bijection φ : V (G)→ V (H) such that G− v ∼= H − φ(v) for each v ∈ V (G). A graph G is reconstructible if H ∼= G for all H hypomorphic to G. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we con...