A power digraph, denoted by $G(n,k)$, is a directed graph with ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\ge 1$ and $k\ge 2$ are determined....

The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes. For any two integers s and n with 1 ≤ s ≤ n, let $D_{n,s}$ be the oriented graph such that $V\left(D_{n,s}\right)$ is the set of integers mod 2n+1 and $A\left(D_{n,s}\right)=(i,j):j-i\in 1,2,...,n\setminus s..$In this paper we prove that $hc\left(D_{n,s}\right)\le 5$ for n ≥ 7. The bound is tight since equality holds when s ∈ n,[(2n+1)/3].

In this paper we investigate the average Horton-Strahler number of all possible tree-structures of binary tries. For that purpose we consider a generalization of extended binary trees where leaves are distinguished in order to represent the location of keys within a corresponding trie. Assuming a uniform distribution for those trees we prove that the expected Horton-Strahler number of a tree with α internal nodes and β leaves that correspond to a key is asymptotically given by $$\frac{4^{2\beta -\alpha }log\left(\alpha \right)(2\beta -1)(\alpha +1)(\alpha +2)\left(\genfrac{}{}{0pt}{}{2\alpha +1}{\alpha -1}\right)}{8\sqrt{\pi }{\alpha }^{3/2}log\left(2\right)(\beta -1)\beta {\left(\genfrac{}{}{0pt}{}{2\beta }{\beta }\right)}^2}$$ provided that α...

In this note, we give an easy and short proof for the theorem by Park and Kim stating that the hypercompetition numbers of hypergraphs with maximum degree at most two is at most two.

In 1968, Vizing conjectured that for any edge chromatic critical graph G = (V,E) with maximum degree △ and independence number α (G), α (G) ≤ [...] . It is known that α (G) < [...] |V |. In this paper we improve this bound when △≥ 4. Our precise result depends on the number n2 of 2-vertices in G, but in particular we prove that α (G) ≤ [...] |V | when △ ≥ 5 and n2 ≤ 2(△− 1)

For a finite group $G$, the intersection graph of $G$ which is denoted by $\Gamma \left(G\right)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H\cap K\ne 1$. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of $\mathrm{Aut}\left(\Gamma \right(G\left)\right)$.

In this paper we first calculate the number of vertices and edges of the intersection graph of ideals of direct product of rings and fields. Then we study Eulerianity and Hamiltonicity in the intersection graph of ideals of direct product of commutative rings.

We study an inverse eigenvalue problem (IEP) of reconstructing a special kind of symmetric acyclic matrices whose graph is a generalized star graph. The problem involves the reconstruction of a matrix by the minimum and maximum eigenvalues of each of its leading principal submatrices. To solve the problem, we use the recurrence relation of characteristic polynomials among leading principal minors. The necessary and sufficient conditions for the solvability of the problem are derived. Finally, a...