[6] **viXra:1906.0502 [pdf]**
*submitted on 2019-06-27 07:21:45*

**Authors:** Daria Grushka, Viktoriia Lebid

**Comments:** 4 Pages. Text in Ukrainian. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152600

Spectral graph theory uses the eigenvalues of matrices associated with a graph to determine the structural properties of the graph. The spectrum of the generalized adjacency matrix is considered in the paper. Graphs with the same spectrum are called cospectral. Is every graph uniquely determined by its spectrum (DS for short)?
This question goes back for about half a century, and originates from chemistry. In 1956 Gunthard and Primas raised the question in a paper that related the theory of graph spectra to Huckel’s theory. At that time it was believed that every graph is determined by the spectrum, until in 1957 Collatz and Sinogowitz presented a pair of cospectral trees. In 1967 Schwenk proved that for almost all trees there is another tree with the same spectrum. Such a statement is neither proved nor refuted for the class of graphs in general. Till now, computational experiments were done on the set of all graphs on up to 12 vertices by Haemers. Computer enumerations for small n show that up to 10 vertices the fraction of graphs that are DS decreases, but for n = 11 and n = 12 it increases again.
We consider the construction of the cospectral graphs called GM-switching for graph G taking the cycle C2n and adjoining a vertex v adjacent to half the vertices of C2n. For these graphs we determine the pairs of cospectral nonisomorphic graphs for small n. It is an operation on graphs that leaves the spectrum of the generalized adjacency matrix invariant. It turns out that for the enumerated cases a large part of all cospectral graphs comes from GM switching, and that the fraction of graphs on n vertices with a cospectral mate starts to decrease at some value of n < 11 (depending on the matrix). Since the fraction of cospectral graphs on n vertices constructible by GM switching tends to 0 if n → ∞, the present data give some indication that possibly almost no graph has a cospectral mate.
Haemers and Spence derived asymptotic lower bounds for the number of graphs with a cospectral mate from GM switching.

**Category:** Combinatorics and Graph Theory

[5] **viXra:1906.0501 [pdf]**
*replaced on 2019-07-22 12:27:42*

**Authors:** Marco Ripà

**Comments:** 12 Pages.

In this paper, we show enhanced upper bounds of the nontrivial n_1 × n_2 × n_3 points problem for every n_1 ≤ n_2 ≤ n_3 < 6. We present new patterns that drastically improve the previously known algorithms for finding minimum-link covering paths, solving completely a few cases (e.g., n_1 = n_2 = 3 and n_3 = 4).

**Category:** Combinatorics and Graph Theory

[4] **viXra:1906.0350 [pdf]**
*submitted on 2019-06-20 03:54:55*

**Authors:** Prajnanaswaroopa S, J Geetha, K Somasundaram

**Comments:** 3 Pages.

In this short note, we give a coloring procedure for graphs which consist of cliques sharing at most one point

**Category:** Combinatorics and Graph Theory

[3] **viXra:1906.0144 [pdf]**
*replaced on 2019-06-10 17:12:23*

**Authors:** Valerio Bencini

**Comments:** 12 Pages.

In this paper, I show an improved upper bound for the case n_1=n_2>n_3 of the n_1 X n_2 X n_3 Dots Puzzle, and I extend all the upper bounds I found to the k-dimensional case, with k>=4.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1906.0110 [pdf]**
*replaced on 2019-07-11 06:37:17*

**Authors:** Valerio Bencini

**Comments:** 8 Pages.

The aim of this paper is to lower down the current upper bound for the Ripà's n_1 X n_2 X n_3 Dots Problem, with n1>n_2>=n_3, using the same method Ripà and I used for the case n_1=n_2=n_3:=n. The new value is 1/2*floor(1/(3*n_1-3*n_2-3*n_3+5))*((n_1-n_2-n_3)*(n_1-n_2-n_3+1)-2*floor(1/2*(-n_1+n_2+n_3)))+2*n_2*n_3-1.
At the end of the article, I also extend this result, and that I previously found with Ripà for n_1=n_2=n_3:=n, to the k-dimensional problem n_1 X n_2 X n_3 X n_4 X ... X n_k, using the equation found by Ripà in 2013.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1906.0031 [pdf]**
*submitted on 2019-06-03 12:10:14*

**Authors:** Henning Thielemann

**Comments:** 5 Pages. Language: German

The dice sequence
The dice sequence is an adaption of Kruskal's card trick to dice.
We compute precise and approximated probabilities that the trick works.
The adaption to dice simplifies the problem considerably
because the probabilities of the dice rolls are independent.
Initially I wrote the text for Wikipedia but in order to meet Wikipedia's exclusion of original research I wrote up this paper.

**Category:** Combinatorics and Graph Theory