Let D 2 $D_2$ denote the 3-uniform hypergraph with 4 vertices and edges. Answering a question of Alon Shapira, we prove an induced removal lemma for having polynomial bounds. We also Erdős–Hajnal-type result: Every -free on n $n$ contains clique or independent set size c $n^{c}$ some absolute constant > 0 $c 0$ . In case both problems, is only nontrivial k $k$ -uniform ⩾ 3 $k\geqslant 3$ which ...

A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, infinitely many non-is...

In this note, we construct bipartite 2-walk-regular graphs with exactly 6 distinct eigenvalues as the point-block incidence graphs of group divisible designs with the dual property. For many of them, we show that they are 2-arc-transitive dihedrants. We note that some of these graphs are not described in Du et al. (2008), in which they classified the connected 2-arc transitive dihedrants.

We generalize three classical characterizations of line graphs to signed and gain graphs: the Krausz's characterization, van Rooij Wilf's characterization Beineke's characterization. In particular, we present a list forbidden subgraphs characterizing class gain-line graphs. case graph whose underlying is graph, this consists exactly four Under same hypothesis, prove that if only its eigenvalues...

A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.

In this paper we study the complementarity spectrum of digraphs, with special attention to problem digraph characterization through spectrum. That is, whether two non-isomorphic digraphs same number vertices can have eigenvalues. The eigenvalues matrices, also called Pareto eigenvalues, has led (undirected) graphs and, in particular, undirected these is an open problem. We characterize one and ...

It is a well-known fact that graph of diameter d has at least d+1 eigenvalues. A d-extremal (resp. dα-extremal) if it and exactly distinct eigenvalues α-eigenvalues), split its vertex set can be partitioned into clique stable set. Such graphs have most three. If all degrees in are either d˜ or d^, then we say (d˜,d^)-bidegreed. In this paper, present complete classification the connected bidegr...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...