# On the extremal values of the number of vertices with an interval spectrum
on the set of proper edge colorings of the graph of the $n$-dimensional cube

A.M. Khachatryan1, R.R. Kamalian2

###### Abstract

For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$
and $E(G)$ the sets of its vertices and edges, respectively. A function
$\varphi:E(G)\rightarrow\\{1,...,t\\}$ is called a proper edge $t$-coloring of
a graph $G$, if adjacent edges are colored differently and each of $t$ colors
is used. The least value of $t$ for which there exists a proper edge
$t$-coloring of a graph $G$ is denoted by $\chi^{\prime}(G)$. For any graph
$G$, and for any integer $t$ satisfying the inequality $\chi^{\prime}(G)\leq
t\leq|E(G)|$, we denote by $\alpha(G,t)$ the set of all proper edge
$t$-colorings of $G$. Let us also define a set $\alpha(G)$ of all proper edge
colorings of a graph $G$:

$\alpha(G)\equiv\bigcup_{t=\chi^{\prime}(G)}^{|E(G)|}\alpha(G,t).$

An arbitrary nonempty finite subset of consecutive integers is called an
interval. If $\varphi\in\alpha(G)$ and $x\in V(G)$, then the set of colors of
edges of $G$ which are incident with $x$ is denoted by $S_{G}(x,\varphi)$ and
is called a spectrum of the vertex $x$ of the graph $G$ at the proper edge
coloring $\varphi$. If $G$ is a graph and $\varphi\in\alpha(G)$, then define
$f_{G}(\varphi)\equiv|\\{x\in V(G)/S_{G}(x,\varphi)\textrm{ is an
interval}\\}|$.

For a graph $G$ and any integer $t$, satisfying the inequality
$\chi^{\prime}(G)\leq t\leq|E(G)|$, we define:

$\mu_{1}(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_{G}(\varphi),\qquad\mu_{2}(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_{G}(\varphi).$

For any graph $G$, we set:

$\mu_{11}(G)\equiv\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t),\qquad\mu_{12}(G)\equiv\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t),$ $\mu_{21}(G)\equiv\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t),\qquad\mu_{22}(G)\equiv\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t).$

For any positive integer $n$, the exact values of the parameters $\mu_{11}$,
$\mu_{12}$, $\mu_{21}$ and $\mu_{22}$ are found for the graph of the
$n$-dimensional cube.

Keywords: $n$-dimensional cube, proper edge coloring, interval spectrum, game.

Math. Classification: 05C15

$\\\ ^{1}$Ijevan Branch of Yerevan State University, e-mail:
<EMAIL_ADDRESS>$\\\ ^{2}$The Institute for Informatics and
Automation Problems of NAS RA,
e-mail<EMAIL_ADDRESS>

We consider finite, undirected, connected graphs without loops and multiple
edges containing at least one edge. For any graph $G$, we denote by $V(G)$ and
$E(G)$ the sets of vertices and edges of $G$, respectively. For any $x\in
V(G)$, $d_{G}(x)$ denotes the degree of the vertex $x$ in $G$. For a graph
$G$, $\delta(G)$ and $\Delta(G)$ denote the minimum and maximum degrees of
vertices in $G$, respectively. For a graph $G$, and for any $V_{0}\subseteq
V(G)$, we denote by $G[V_{0}]$ the subgraph of the graph $G$ induced [1] by
the subset $V_{0}$ of its vertices. For arbitrary graphs $G$ and $H$, $G\Box
H$ denotes their cartesian product [1].

For any positive integer $n$, we denote by $Q_{n}$ the graph of the
$n$-dimensional cube [2]. Clearly, for any positive integer $n$, $Q_{n}$ is a
regular bipartite graph with $|V(Q_{n})|=2^{n}$, $|E(Q_{n})|=n\cdot 2^{n-1}$,
$\Delta(Q_{n})=n$.

An arbitrary nonempty finite subset of consecutive integers is called an
interval. An interval with the minimum element $p$ and the maximum element $q$
is denoted by $[p,q]$.

A function $\varphi:E(G)\rightarrow[1,t]$ is called a proper edge $t$-coloring
of a graph $G$, if each of $t$ colors is used, and adjacent edges are colored
differently.

The minimum value of $t$ for which there exists a proper edge $t$-coloring of
a graph $G$ is denoted by $\chi^{\prime}(G)$ [3].

For any graph $G$, and for any $t\in[\chi^{\prime}(G),|E(G)|]$, we denote by
$\alpha(G,t)$ the set of all proper edge $t$-colorings of $G$.

Let us also define a set $\alpha(G)$ of all proper edge colorings of a graph
$G$:

$\alpha(G)\equiv\bigcup_{t=\chi^{\prime}(G)}^{|E(G)|}\alpha(G,t).$

If $\varphi\in\alpha(G)$ and $x\in V(G)$, then the set $\\{\varphi(e)/e\in
E(G),e\textrm{ is incident with }x$} is called a spectrum of the vertex $x$ of
the graph $G$ at the proper edge coloring $\varphi$ and is denoted by
$S_{G}(x,\varphi)$; if $S_{G}(x,\varphi)$ is an interval, we say that
$\varphi$ is interval in $x$.

If $G$ is a graph, $\varphi\in\alpha(G)$, $R\subseteq V(G)$, then we say, that
$\varphi$ is interval on $R$ iff for $\forall x\in R$, $\varphi$ is interval
in $x$. We say, that a subset $R$ of vertices of a graph $G$ has an
$i$-property iff there exists $\varphi\in\alpha(G)$ interval on $R$. If $G$ is
a graph, and a subset $R$ of its vertices has an $i$-property, we denote by
$w_{R}(G)$ and $W_{R}(G)$ (omiting the index in these notations in the
peculiar case of $R=V(G)$) the minimum and the maximum value of $t$,
respectively, for which $\exists\varphi\in\alpha(G,t)$ interval on $R$ [4, 5,
6] (see also [20]).

If $G$ is a graph, $\varphi\in\alpha(G)$, then set
$V_{int}(G,\varphi)\equiv\\{x\in V(G)/S_{G}(x,\varphi)\textrm{ is an
interval}\\}$ and $f_{G}(\varphi)\equiv|V_{int}(G,\varphi)|$. A proper edge
coloring $\varphi\in\alpha(G)$ is called an interval edge coloring [4, 5, 6]
of the graph $G$ iff $f_{G}(\varphi)=|V(G)|$. The set of all graphs having an
interval edge coloring is denoted by $\mathfrak{N}$. The simplest example of
the graph which doesn’t belong to $\mathfrak{N}$ is $K_{3}$. The terms and
concepts which are not defined can be found in [1].

For a graph $G$, and for any $t\in[\chi^{\prime}(G),|E(G)|]$, we set [7]:

$\mu_{1}(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_{G}(\varphi),\qquad\mu_{2}(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_{G}(\varphi).$

For any graph $G$, we set [7]:

$\mu_{11}(G)\equiv\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t),\qquad\mu_{12}(G)\equiv\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t),$ $\mu_{21}(G)\equiv\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t),\qquad\mu_{22}(G)\equiv\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t).$

Clearly, the parameters $\mu_{11}$, $\mu_{12}$, $\mu_{21}$ and $\mu_{22}$ are
correctly defined for an arbitrary graph.

Let us note that exact values of the parameters $\mu_{12}$ and $\mu_{21}$ have
certain game interpretations. Suppose that all edges of a graph $G$ are
colored in the game of Alice and Bob with antagonistic interests and
asymmetric distribution of roles. Alice determines the number $t$ of colors in
the future proper edge coloring $\varphi$ of the graph $G$, satisfying the
condition $t\in[\chi^{\prime}(G),|E(G)|]$, Bob colors edges of $G$ with $t$
colors.

When Alice aspires to maximize, Bob aspires to minimize the value of the
function $f_{G}(\varphi)$, and both players choose their best strategies, then
at the finish of the game exactly $\mu_{12}(G)$ vertices of $G$ will receive
an interval spectrum.

When Alice aspires to minimize, Bob aspires to maximize the value of the
function $f_{G}(\varphi)$, and both players choose their best strategies, then
at the finish of the game exactly $\mu_{21}(G)$ vertices of $G$ will receive
an interval spectrum.

The exact values of the parameters $\mu_{11}$, $\mu_{12}$, $\mu_{21}$ and
$\mu_{22}$ are found for simple paths, simple cycles and simple cycles with a
chord [8, 9], "Möbius ladders" [7, 22], complete graphs [11], complete
bipartite graphs [12, 13], prisms [14, 22], $n$-dimensional cubes [14, 15] and
the Petersen graph [23]. The exact values of $\mu_{11}$ and $\mu_{22}$ for
trees are found in [16]. The exact value of $\mu_{12}$ for an arbitrary tree
is found in [17] (see also [18, 19]).

In this paper, for any positive integer $n$, we determine the exact values of
the parameters $\mu_{11}$, $\mu_{12}$, $\mu_{21}$ and $\mu_{22}$ for the graph
$Q_{n}$.

First we recall some known results.

######  Lemma 1.

[4, 5, 6] If $G\in\mathfrak{N}$, then $\chi^{\prime}(G)=\Delta(G)$.

######  Corollary 1.

If $G\in\mathfrak{N}$, then $\Delta(G)=\chi^{\prime}(G)\leq w(G)\leq
W(G)\leq|E(G)|$.

######  Corollary 2.

If $G$ is a regular graph, then $G\in\mathfrak{N}$ iff
$\chi^{\prime}(G)=\Delta(G)$.

######  Corollary 3.

If $G$ is a regular graph with $\chi^{\prime}(G)=\Delta(G)$, then
$G\in\mathfrak{N}$ and $w(G)=\chi^{\prime}(G)=\Delta(G)$.

######  Corollary 4.

If $G$ is a regular bipartite graph, then $G\in\mathfrak{N}$ and
$w(G)=\chi^{\prime}(G)=\Delta(G)$.

######  Corollary 5.

If $G\in\mathfrak{N}$, and a subset $R\subseteq V(G)$ has an $i$-property,
then $\Delta(G)=\chi^{\prime}(G)\leq w_{R}(G)\leq w(G)\leq W(G)\leq
W_{R}(G)\leq|E(G)|$.

######  Lemma 2.

[4, 5, 6] If $R$ is the set of all vertices of an arbitrary part of a
bipartite graph $G$, then:

  1. 1)

$R$ has an $i$-property,

  2. 2)

$W_{R}(G)=|E(G)|$,

  3. 3)

for any $t\in[w_{R}(G),W_{R}(G)]$, there exists $\varphi_{t}\in\alpha(G,t)$
interval on $R$.

######  Corollary 6.

If $R$ is the set of all vertices of an arbitrary part of a regular bipartite
graph $G$, then:

  1. 1)

$R$ has an $i$-property,

  2. 2)

$W_{R}(G)=|E(G)|$,

  3. 3)

$w_{R}(G)=\Delta(G)$,

  4. 4)

for any $t\in[w_{R}(G),W_{R}(G)]$, there exists $\varphi_{t}\in\alpha(G,t)$
interval on $R$.

######  Lemma 3.

[21] If $G$ is a graph with $\delta(G)\geq 2$, $\varphi\in\alpha(G,|E(G)|)$,
$V_{int}(G,\varphi)\neq\emptyset$, then $G[V_{int}(G,\varphi)]$ is a forest
each connected component of which is a simple path.

######  Lemma 4.

[8, 9] If $G$ is a regular graph with $\chi^{\prime}(G)=\Delta(G)$, then
$\mu_{12}(G)=|V(G)|$.

Proof.  Let us note that for an arbitrary $\varphi\in\alpha(G,\Delta(G))$,
$f_{G}(\varphi)=|V(G)|$. Consequently,
$|V(G)|=\mu_{1}(G,\Delta(G))\leq\max_{t\in[\chi^{\prime}(G),|E(G)|]}\mu_{1}(G,t)=\mu_{12}(G)\leq|V(G)|$.
Hence, $\mu_{12}(G)=|V(G)|$.

The Lemma is proved.

######  Lemma 5.

[8, 9] For an arbitrary graph $G$, the following inequalities hold:
$\mu_{11}(G)\leq\mu_{12}(G)\leq\mu_{22}(G)$,
$\mu_{11}(G)\leq\mu_{21}(G)\leq\mu_{22}(G)$.

Proof.  Clearly,

$\mu_{11}(G)=\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t)\leq\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t)=\mu_{12}(G).$

and

$\mu_{21}(G)=\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t)\leq\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t)=\mu_{22}(G).$

The inequality $\mu_{12}(G)\leq\mu_{22}(G)$ is provided by the existence of an
integer $t^{\prime}\in[\chi^{\prime}(G),|E(G)|]$, for which the relation

$\mu_{12}(G)=\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t)=\mu_{1}(G,t^{\prime})=\min_{\varphi\in\alpha(G,t^{\prime})}f_{G}(\varphi)\leq$
$\leq\max_{\varphi\in\alpha(G,t^{\prime})}f_{G}(\varphi)=\mu_{2}(G,t^{\prime})\leq\max_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t)=\mu_{22}(G)$

is true.

The inequality $\mu_{11}(G)\leq\mu_{21}(G)$ is provided by the existence of an
integer $t^{\prime\prime}\in[\chi^{\prime}(G),|E(G)|]$, for which the relation

$\mu_{21}(G)=\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{2}(G,t)=\mu_{2}(G,t^{\prime\prime})=\max_{\varphi\in\alpha(G,t^{\prime\prime})}f_{G}(\varphi)\geq$
$\geq\min_{\varphi\in\alpha(G,t^{\prime\prime})}f_{G}(\varphi)=\mu_{1}(G,t^{\prime\prime})\geq\min_{\chi^{\prime}(G)\leq
t\leq|E(G)|}\mu_{1}(G,t)=\mu_{11}(G)$

is true.

The Lemma is proved.

######  Remark 1.

[8, 9] There are graphs $G$ with $\mu_{21}(G)<\mu_{12}(G)$. There are graphs
$G$ with $\mu_{12}(G)<\mu_{21}(G)$.

######  Lemma 6.

[10, 24] If a subset $V_{0}$ of the set of vertices of the graph $Q_{3}$
contains at least $6$ vertices, then at least one of the following two
statements is true:

  1. 1)

there exist such vertices $x_{1},x_{2},x_{3},x_{4}$ in $V_{0}$ that
$Q_{3}[\\{x_{1},x_{2},x_{3},x_{4}\\}]\cong K_{3,1}$,

  2. 2)

there exist such vertices $y_{1},y_{2},y_{3},y_{4},y_{5},y_{6}$ in $V_{0}$
that $Q_{3}[\\{y_{1},y_{2},y_{3},y_{4},y_{5},y_{6}\\}]\cong C_{6}$.

######  Lemma 7.

[10, 24] If a subset $V_{0}$ of the set of vertices of the graph $Q_{n}$
$(n\geq 4)$ contains at least $2^{n-1}+1$ vertices, then at least one of the
following two statements is true:

  1. 1)

there exist such vertices $x_{1},x_{2},x_{3},x_{4}$ in $V_{0}$ that
$Q_{n}[\\{x_{1},x_{2},x_{3},x_{4}\\}]\cong K_{3,1}$,

  2. 2)

there exist such vertices $y_{1},y_{2},y_{3},y_{4},y_{5},y_{6},y_{7},y_{8}$ in
$V_{0}$ that
$Q_{n}[\\{y_{1},y_{2},y_{3},y_{4},y_{5},y_{6},y_{7},y_{8}\\}]\cong C_{8}$.

If $G$ is a graph with $\chi^{\prime}(G)=\Delta(G)$, $t\in[\Delta(G),|E(G)|]$,
$\xi\in\alpha(G,t)$, then for any $j\in[1,t]$, we denote by $E(G,\xi,j)$ the
set of all edges of $G$ colored by the color $j$ at the proper edge
$t$-coloring $\xi$; $\xi$ is called a harmonic [12] edge $t$-coloring of the
graph $G$, if for any $i\in[1,\Delta(G)]$, the set

$\bigcup_{1\leq j\leq t,\;j\equiv i(mod(\Delta(G)))}E(G,\xi,j)$

is a matching in $G$.

Suppose that $G$ is a graph with $\chi^{\prime}(G)=\Delta(G)$,
$t\in[\Delta(G),|E(G)|]$, $\xi$ is a harmonic edge $t$-coloring of $G$. Let us
define a sequence
$\xi^{*}_{0},\xi^{*}_{1},\ldots,\xi^{*}_{t-\chi^{\prime}(G)}$ of proper edge
colorings of the graph $G$.

Set $\xi^{*}_{0}\equiv\xi$.

Case $\mathbf{1}$. $t=\chi^{\prime}(G)$. The desired sequence mentioned above
is already constructed.

Case $\mathbf{2}$. $t\in[\chi^{\prime}(G)+1,|E(G)|]$. Suppose that
$j\in[1,t-\chi^{\prime}(G)]$, and the proper edge colorings
$\xi^{*}_{0},\ldots,\xi^{*}_{j-1}$ of the graph $G$ are already constructed.
Let us define $\xi^{*}_{j}$. For an arbitrary $e\in E(G)$, set:

$\xi^{*}_{j}(e)\equiv\left\\{\begin{array}[]{ll}\xi^{*}_{j-1}(e),&\textrm{if
$\xi^{*}_{j-1}(e)\neq\max(\\{\xi^{*}_{j-1}(e)/e\in E(G)\\})$}\\\
\xi^{*}_{j-1}(e)-\Delta(G),&\textrm{if
$\xi^{*}_{j-1}(e)=\max(\\{\xi^{*}_{j-1}(e)/e\in E(G)\\})$}.\end{array}\right.$

######  Remark 2.

Suppose that $G$ is a graph with $\chi^{\prime}(G)=\Delta(G)$,
$t\in[\Delta(G),|E(G)|]$, $\xi$ is a harmonic edge $t$-coloring of $G$. All
proper edge colorings
$\xi^{*}_{0},\xi^{*}_{1},\ldots,\xi^{*}_{t-\chi^{\prime}(G)}$ of the graph $G$
are determined definitely.

######  Remark 3.

Suppose that $G$ is a graph with $\chi^{\prime}(G)=\Delta(G)<|E(G)|$,
$t\in[1+\chi^{\prime}(G),|E(G)|]$, $\xi$ is a harmonic edge $t$-coloring of
$G$. It is not difficult to see, that for any $j\in[1,t-\chi^{\prime}(G)]$,
$\xi^{*}_{j}$ is a harmonic edge $(t-j)$-coloring of the graph $G$.

######  Remark 4.

Suppose that $G$ is a graph with $\chi^{\prime}(G)=\Delta(G)<|E(G)|$,
$t\in[1+\chi^{\prime}(G),|E(G)|]$, $\xi$ is a harmonic edge $t$-coloring of
$G$. Assume that $\xi$ is interval in some vertex $z_{0}\in V(G)$ with
$d_{G}(z_{0})=\Delta(G)$. Then, for any $j\in[1,t-\chi^{\prime}(G)]$,
$\xi^{*}_{j}$ is interval in $z_{0}$.

From results of [8, 9] we have

######  Proposition 1.

$\mu_{11}(Q_{1})=2$, $\mu_{11}(Q_{2})=1$, $\mu_{21}(Q_{1})=2$,
$\mu_{21}(Q_{2})=3$.

######  Lemma 8.

If an $r$-regular $(r\geq 3)$ graph $G$ has $\varphi\in\alpha(G,r+1)$ with
$f_{G}(\varphi)=0$, then the graph $G\Box K_{2}$ has $\psi\in\alpha(G\Box
K_{2},r+2)$ with $f_{G\Box K_{2}}(\psi)=0$.

Proof is evident.

######  Lemma 9.

There exists $\varphi\in\alpha(Q_{3},4)$ with $f_{Q_{3}}(\varphi)=0$. There
exists $\psi\in\alpha(Q_{3},12)$ with $f_{Q_{3}}(\psi)=5$.

Proof.  Let $V(Q_{3})=\\{x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4}\\}$,
$E(Q_{3})=\\{(x_{1},x_{2}),(x_{2},x_{3}),(x_{3},x_{4}),(x_{1},x_{4}),\\\
(y_{1},y_{2}),(y_{2},y_{3}),(y_{3},y_{4}),(y_{1},y_{4}),(x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3}),(x_{4},y_{4})\\}$.

Set:
$\varphi((x_{1},x_{2}))=\varphi((x_{3},x_{4}))=\varphi((y_{1},y_{2}))=\varphi((y_{3},y_{4}))=1$,
$\varphi((x_{1},x_{4}))=\varphi((x_{2},x_{3}))=2$,
$\varphi((y_{1},y_{4}))=\varphi((y_{2},y_{3}))=3$,
$\varphi((x_{1},y_{1}))=\varphi((x_{2},y_{2}))=\varphi((x_{3},y_{3}))=\varphi((x_{4},y_{4}))=4$.

It is not difficult to see that $\varphi\in\alpha(Q_{3},4)$ and
$f_{Q_{3}}(\varphi)=0$.

Set: $\psi((x_{1},y_{1}))=1$, $\psi((y_{1},y_{2}))=2$,
$\psi((y_{1},y_{4}))=3$, $\psi((y_{3},y_{4}))=4$, $\psi((x_{4},y_{4}))=5$,
$\psi((x_{1},x_{4}))=6$, $\psi((x_{3},x_{4}))=7$, $\psi((x_{3},y_{3}))=8$,
$\psi((x_{2},x_{3}))=9$, $\psi((x_{2},y_{2}))=10$, $\psi((x_{1},x_{2}))=11$,
$\psi((y_{2},y_{3}))=12$.

It is not difficult to see that $\psi\in\alpha(Q_{3},12)$ and
$f_{Q_{3}}(\psi)=5$.

The Lemma is proved.

From Lemmas 8 and 9 we obtain

######  Proposition 2.

For any integer $n\geq 3$, $\mu_{11}(Q_{n})=0$.

From Propositions 1 and 2 we obtain

######  Proposition 3.

For any positive integer $n$, $\mu_{11}(Q_{n})=3-\min\\{3,n\\}$.

From Lemma 4 we obtain

######  Proposition 4.

For any positive integer $n$, $\mu_{12}(Q_{n})=2^{n}$.

From Proposition 4 and Lemma 5 we obtain

######  Proposition 5.

For any positive integer $n$, $\mu_{22}(Q_{n})=2^{n}$.

######  Lemma 10.

For any integer $n\geq 4$, $\mu_{2}(Q_{n},n\cdot 2^{n-1})\geq 2^{n-1}$.

Proof follows from Lemma 2.

######  Lemma 11.

For any integer $n\geq 4$, $\mu_{2}(Q_{n},n\cdot 2^{n-1})\leq 2^{n-1}$.

Proof.  Assume the contrary. Then there exist an integer $n_{0}\geq 4$ and
$\varphi_{0}\in\alpha(Q_{n_{0}},n_{0}\cdot 2^{n_{0}-1})$, for which
$f_{Q_{n_{0}}}(\varphi_{0})\geq 2^{n_{0}-1}+1$. By Lemma 3,
$Q_{n_{0}}[V_{int}(Q_{n_{0}},\varphi_{0})]$ is a forest, each connected
component of which is a simple path. But it is incompatible with Lemma 7.

The Lemma is proved.

From Lemmas 10 and 11 we obtain

######  Lemma 12.

For any integer $n\geq 4$, $\mu_{2}(Q_{n},n\cdot 2^{n-1})=2^{n-1}$.

######  Lemma 13.

For arbitrary integers $n$ and $t$, satisfying the conditions $n\geq 4$,
$n\leq t\leq n\cdot 2^{n-1}$, the inequality $\mu_{2}(Q_{n},t)\geq 2^{n-1}$ is
true.

Proof.  Choose an arbitrary integer $n_{0}\geq 4$. It follows from Corollary
6, that for any integer $t$, satisfying the inequality $n_{0}\leq t\leq
n_{0}\cdot 2^{n_{0}-1}$, there exists $\varphi_{t}\in\alpha(Q_{n_{0}},t)$ with
$f_{Q_{n_{0}}}(\varphi_{t})\geq 2^{n_{0}-1}$. It means that for any integer
$t$, satisfying the inequality $n_{0}\leq t\leq n_{0}\cdot 2^{n_{0}-1}$, we
also have $\mu_{2}(Q_{n_{0}},t)\geq 2^{n_{0}-1}$.

The Lemma is proved.

From Lemmas 12 and 13 we obtain

######  Proposition 6.

For any integer $n\geq 4$, $\mu_{21}(Q_{n})=2^{n-1}$.

From Lemma 9 we obtain

######  Lemma 14.

$\mu_{2}(Q_{3},12)\geq 5$.

######  Lemma 15.

$\mu_{2}(Q_{3},12)\leq 5$.

Proof.  Assume the contrary: $\mu_{2}(Q_{3},12)\geq 6$. It means that there
exists $\xi\in\alpha(Q_{3},12)$ with $f_{Q_{3}}(\xi)\geq 6$. By Lemma 3,
$Q_{3}[V_{int}(Q_{3},\xi)]$ is a forest each connected component of which is a
simple path. But it is incompatible with Lemma 6.

The Lemma is proved.

From Lemmas 14 and 15 we obtain

######  Lemma 16.

$\mu_{2}(Q_{3},12)=5$.

######  Lemma 17.

For any integer $t$, satisfying the condition $3\leq t\leq 12$, the inequality
$\mu_{2}(Q_{3},t)\geq 5$ is true.

Proof.  It is not difficult to check that the proper edge $12$-coloring $\psi$
of the graph $Q_{3}$ constructed for the proof of Lemma 9 is a harmonic edge
$12$-coloring of $Q_{3}$. From Remark 4 it follows that for any integer $t$
satisfying the inequality $3\leq t\leq 11$, there exists
$\psi_{t}\in\alpha(Q_{3},t)$ with $V_{int}(Q_{3},\psi)\subseteq
V_{int}(Q_{3},\psi_{t})$. Hence, for the same values of $t$, the inequality
$\mu_{2}(Q_{3},t)\geq f_{Q_{3}}(\psi)=5$ is true.

The Lemma is proved.

From Lemmas 16 and 17 we obtain

######  Proposition 7.

$\mu_{21}(Q_{3})=5$.

From Propositions 1, 6 and 7 we obtain

######  Proposition 8.

For any positive integer $n$,

$\mu_{21}(Q_{n})=2^{n-1}+\bigg{\lceil}1-\frac{\min\\{4,n\\}}{4}\bigg{\rceil}.$

From Propositions 3 \- 5 and 8 we obtain

######  Theorem 1.

[14, 15] For any positive integer $n$, the following equalities hold:

$\mu_{11}(Q_{n})=3-\min\\{3,n\\},\qquad\mu_{12}(Q_{n})=\mu_{22}(Q_{n})=2^{n},$
$\mu_{21}(Q_{n})=2^{n-1}+\bigg{\lceil}1-\frac{\min\\{4,n\\}}{4}\bigg{\rceil}.$

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  * [6] R.R. Kamalian, Interval Edge Colorings of Graphs, Doctoral dissertation, the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of USSR, Novosibirsk, 1990 (in Russian).
  * [7] N.N. Davtyan, R.R. Kamalian, On boundaries of extremums of the number of vertices with an interval spectrum among the set of proper edge colorings of "Möbius ladders" with $t$ colors under variation of $t$, Proc. of the $3^{th}$ Ann. Sci. Conf. (December 5–10, 2008) of the RAU, Yerevan, 2009, pp. 81–84 (in Russian).
  * [8] N.N. Davtyan, R.R. Kamalian, On properties of the number of vertices with an interval spectrum in proper edge colorings of some graphs, the Herald of the RAU, 2, Yerevan, 2009, pp. 33–42.
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  * [10] R.R. Kamalian, A.M. Khachatryan, On a property of the $n$-dimensional cube, Mathematical Problems of Computer Science, Vol. 38, Yerevan, 2012, pp. 66–67.
  * [11] A.M. Khachatryan, On boundaries of extremums of the number of vertices with an interval spectrum among the set of proper edge colorings of complete graphs with $t$ colors under variation of $t$, Proc. of the $5^{th}$ Ann. Sci. Conf. (December 6–10, 2010) of the RAU, Yerevan, 2011, pp. 268–272 (in Russian).
  * [12] A.M. Khachatryan, On the parameters $\mu_{11}$, $\mu_{12}$ and $\mu_{22}$ of complete bipartite graphs, the Herald of the RAU, 1, Yerevan, 2011, pp. 76–83 (in Russian).
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11institutetext: Swarthmore College, Swarthmore PA 19081, USA 11email:
<EMAIL_ADDRESS>22institutetext: Haverford College,
Haverford, PA 19041, USA 22email<EMAIL_ADDRESS>
# Trends in Machine Learning and Electroencephalogram (EEG): A Review for
Undergraduate Researchers

Nathan Koome Murungi 11 0009-0008-3170-7689 Michael Vinh Pham 11
0009-0007-5322-1543 Xufeng Dai 22 0009-0006-0513-1255 Xiaodong Qu Nathan,
Michael, and Xufeng are the first three authors of this paper, and they
contributed equally. Professor Xiaodong Qu is the mentor for this research
project.11 0000-0001-7610-6475

###### Abstract

This paper presents a systematic literature review on Brain-Computer
Interfaces (BCIs) in the context of Machine Learning. Our focus is on
Electroencephalography (EEG) research, highlighting the latest trends as of
2023. The objective is to provide undergraduate researchers with an accessible
overview of the BCI field, covering tasks, algorithms, and datasets. By
synthesizing recent findings, our aim is to offer a fundamental understanding
of BCI research, identifying promising avenues for future investigations.

###### Keywords:

Machine Learning Deep Learning Brain-Computer Interfaces BCI
Electroencephalography EEG Undergrad Review

##  1 Introduction

Since the advent of computing, the disparity between human and computer
technology has significantly diminished. Starting with early human-computer
interfaces like keyboards and microphones, the boundary between humans and
computers has been progressively blurred, primarily owing to the emergence and
utilization of brain-computer interfaces [34]. In the rapidly advancing field
of Brain-Computer Interfaces (BCI), Electroencephalography (EEG) analysis
plays a crucial role in establishing a connection between the human brain and
Machine Learning (ML) algorithms [5, 11, 37, 40, 43, 44, 45]. The
proliferation of ML algorithms and the increasing availability of EEG data
have created exciting opportunities for researchers to explore new approaches
to interpreting raw EEG data. However, this progress presents a challenge for
newcomers due to the overwhelming volume of research papers and the rapid rate
at which they become outdated, making it challenging to navigate the research
landscape effectively.

To tackle this formidable challenge, we present a meticulous examination of
the existing literature in the field of Brain-Computer Interfaces (BCI), with
a specific focus on the most recent advancements up to 2023. This paper is
tailored to cater to undergraduate researchers, serving as a comprehensive
overview and guide for those aspiring to conduct research in the
Electroencephalography (EEG) domain. By systematically analyzing and
organizing the obtained findings, our objective is to facilitate a profound
comprehension of the current landscape of BCI research while identifying
promising avenues for future investigations.

In addition to providing a comprehensive review, this paper specifically
delves into utilizing Transformers, a prominent and rapidly emerging machine
learning algorithm, within the realm of BCI research. By focusing on the
application of Transformers in this context, we aim to shed light on its
significance and impact, offering insights into its potential advantages and
limitations. Table 1 lists the acronyms used in this paper.

Abbreviation | Definition
---|---
ML | Machine Learning
DL | Deep Learning
LSTM | Long Short-Term Memory
CNN | Convolutional Neural Network
DNN | Deep Neural Network
AE | Autoencoder
GAN | Generative Adversarial Network
SVM | Support Vector Machine
RNN | Recurrent Neural Network
ANN | Artificial Neural Network
RF | Random Forest
KNN | K-Nearest Neighbor
DBN | Deep Belief Network
EEG | Electroencephalography
BCI | Brain-Computer Interfaces
Table 1: List of Acronyms

###  1.1 Research Questions

Our research aims to address the following questions at the intersection of ML
and EEG research:

  * •

What are the most suitable tasks, datasets, and ML algorithms for
undergraduate researchers to explore the realms of Machine Learning (ML) and
Electroencephalography (EEG)?

  * •

What are the prevailing trends observed within the intersection of ML and EEG
in 2023?

By answering the first question, we aim to provide guidance to undergraduate
researchers, helping them identify appropriate starting points and resources
for their ML and EEG investigations. This will enable a seamless entry into
the research domain and facilitate their understanding of fundamental concepts
and methodologies.

Regarding the second question, we plan to provide an overview of the current
trends within the EEG-ML field, highlighting the latest advancements and
emerging research directions as of 2023. Understanding these trends may help
undergrad researchers stay informed about cutting-edge developments and
identify promising avenues for their investigations.

##  2 Methods

Figure 1: Selection process for the papers

###  2.1 Keywords

Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) is
the systematic review method that was used to identify relevant EEG research
papers that use ML methods, as mentioned in [5, 43]. This search was conducted
in February 2023 within Google Scholar, Paperwithcode, arXiv, and PubMed
databases using keywords: (’Machine Learning’ OR ’LSTM’ OR ’Deep Learning’ OR
’CNN’ OR ’RNN’ OR ’Classification’ OR ’DNN’ OR ’Autoencoder’ OR ’GAN’ OR ’
Transformer’ AND ’Brain Computer Interface’ OR ’Motor Imagery Classification’
OR ’seizure’ OR ’EEG Review’ OR ’Speech’ OR ’Emotion’ OR ’Parkinson’s’ OR
’Alzheimer’s’ OR ’Depression’ OR ’Rehabilitation’ OR ’Gender Classification’
OR ’Stroke’ OR ’Person Identification’ OR ’Age Classification’ OR ’Task
classification’) AND (’EEG’ OR ’Electroencephalography’ AND ’Survey’ or
’Review’). Figure 1 visualizes this screening method and shows how the best 76
papers were selected and further narrowed down to the 9 recommended papers for
undergraduates, as they are limited in time. Furthermore, with the
understanding that undergraduate researchers are limited in time, we also
narrowed all the papers we reviewed down to 9 suggested research papers that a
new undergraduate researcher should start with and read to get acclimated with
the current trends within the field.

The criteria for selecting appropriate papers are as follows:

  * •

EEG only:  We only look at Electroencephalography (EEG) studies with human
subjects to keep type of data consistent.

  * •

Time: Due to constantly new discoveries within the Machine Learning field and
EEG field, we only included up-to-date literature published in 2020 and after.

  * •

ML/DL included: Highly focused on Machine Learning and Deep Learning
approaches to EEG processing.

  * •

Reproducibility: Papers that include data preprocessing, feature extraction,
results, code, and data source.

  * •

Target Audience: Comprehensible to undergraduate students majoring or minoring
in computer science, interested in the topic of machine learning.

##  3 Results

###  3.1 Tasks

In the healthcare field, EEG data can be decoded to understand the underlying
and psychological status of affected individuals[2, 39].

From our review, most BCI research has been conducted in the clinical setting,
specifically MI, Seizure, and Emotion Detection. As such, and due to their
high accuracy, it’s recommended that undergraduates start with these tasks.
Table 2 shows the most common tasks found for BCI research (with the top 3
bolded). Also there are nonclinical tasks, such as [41, 55, 63] Figure 2 shows
the most common algorithms used for ML. The most common algorithms for Emotion
were CNN, RNN, SVM, and KNN. For Seizure, the most commonly used were CNN,
RNN, Transformer, and KNN. These top 3 tasks found in our literature review
are described below.

####  3.1.1 Motor Imagery

Motor imagery EEG tasks involve the measurement and analysis of EEG brain
activity patterns associated with mentally envisioning particular motor
actions. Rehabilitation and assistive technologies can benefit from EEG/ML
Motor Imagery by facilitating the development of solutions for individuals
with motor and limb impairments via a means of communication from acquired
neural activity of the kinesthetic imagination of limbs. [48] Some healthcare
systems utilize EEG signals to allow individuals to sense and interact with
the physical world by controlling exoskeletons, wheelchairs, and other
assistive technologies [2].

Not only can patients with motor disabilities benefit from EEG/ML Motor
Imagery, but also everyday users through the development of Brain-Computer
Interfaces to control specific motor actions. Current work done in classifying
imaginations of the right or left hand and fingers for example could translate
to a future where users are able to imagine specific motor actions such as
moving a cursor, controlling a robotic arm, or navigating a video game. [2]

####  3.1.2 Emotion Recognition

Emotion detection EEG/ML tasks entail identifying and classifying patterns of
EEG brain activity associated with different emotional states[57]. Emotion is
commonly associated with perception, decision-making, and human interaction.
As the BCI field continues to grow, the interest in establishing an
”emotional” interaction between humans and computers has increased in consumer
products such as virtual reality [50]. In addition, a persons mental condition
and emotional state can be apparent through their EEG waves. Therefore,
emotion detection finds application in mental health contexts, enabling remote
assessment and monitoring of patients’ emotional states by therapists or
counselors which can facilitate timely interventions and support and better
treatment. Emotion recognition is the leading scientific problem in Affective
Computing, which is how computer systems recognize and comprehend emotional
information for natural human-computer interactions. With better Affective
Computing, AI chatbots and voice assistants can better understand human users’
emotions providing more personalized and empathetic interactions. [25]

####  3.1.3 Seizure Detection

Epileptic seizures are chronic neurological diseases that can substantially
impact the lifestyle of an affected individual. For some patients, there could
be hundreds of seizures a day, which greatly affects their brain. They are
sudden abnormalities in the brain’s electrical activities, which is manifested
as excessive discharge of the neuronal network in the cerebral cortex [1].
Accurate and timely detection for seizures can greatly help improve the
livelihood of many affected individuals by preemptively describing seizure
risk which can allow an individual to seek medical expertise. For studies that
highlight seizure detection, EEG signals are recorded during the phases of a
seizure: prodromal, early ictal, ictal, and postictal as well as seizure free
periods. In addition, the control class consisted of non-epileptic subjects.

Task | Paper Count
---|---
Motor Imagery | 19
Seizure | 12
Emotion | 12
Parkinson’s | 3
Gender/Age | 2
Depression | 2
Stroke | 1
Person Identification | 1
Inner Speech | 1
Dementia | 1
Lie Detection | 1
Table 2: Task Breakdown for Non-Review Papers
CNNRNNTransformerSVMANN$0$$5$$10$$15$$12$$2$$6$$1$$1$Number of
OccurrencesMotor Imagery Algorithms Figure 2: Breakdown of Algorithms used for
Motor Imagery Classification

###  3.2 Algorithms

Compared with review papers in previous years [5, 9, 38, 43, 61, 62], many of
the algorithm findings were the same such as CNN dominating as the most
frequently used in BCI research. But there were also interesting trends with
Transformers and DBN. Transformers, a new ML algorithm, replace RNNs with
sequential data because they have a self-attention mechanism that allows them
to process input sequences of varying lengths and extract relevant information
from them [56]. DBN also makes this list as an algorithm that has fallen out
of trend. Despite making up a sizeable portion of earlier BCI research and
review, rarely any research, we came across used DBN. Table 3 shows the most
common algorithms found for BCI research.

Algorithm | Paper Count
---|---
CNN | 20
RNN | 13
Transformer | 5
SVM | 4
KNN | 4
ANN | 4
RF | 3
AE | 2
Table 3: Algorithm Breakdown for Non-Review Papers

###  3.3 Datasets

The datasets used in the studies vary based on the task that the study focuses
on, and the most used are BCI Competition Dataset IV for validating signal
processing and classification method; DEAP for analyzing human affect states;
SEED, a various use EEG dataset; and EEGEyeNet, and Eye-tracking Dataset and
Benchmark for Eye Movement Prediction. Other available datasets include
‘Thinking out loud,’ an EEG-based BCI dataset for inner speech recognition,
‘Feeling Emotions,’ and EEG Brainwave Dataset for emotion recognition. The
datasets cover a wide range of topics for BCI research and are accessible to
undergraduate students.

Dataset | Task | Year | Cited
---|---|---|---
DEAP [21] | Emotion | 2011 | 3439
BCI Competition IV [53] | MI | 2012 | 783
Dreamer [20] | Emotion | 2016 | 517
SEED [10] | Emotion | 2013 | 358
Bonn [24] | Seizure | 2013 | 316
CHB-MIT [36] | Seizure | 2021 | 21
EEGEyeNet [19] | MI | 2021 | 16
Table 4: Dataset Breakdown for Non-Review Papers

##  4 Discussion

We identified the most recent trend in Machine learning BCI research based on
the papers we reviewed. We presented our recommendations for the top three
tasks, algorithms, datasets, and review papers for undergraduate researchers.

It is clear that transformers are becoming more and more utilized in the last
year to classify EEG data. Transformers have traditionally been used in
natural language processing (NLP) due to their ability to handle long-range
dependencies [51]. Therefore, transformers can be efficient with the
recognition of EEG data, a set of time series signals. However, transformers
are commonly paired with algorithms such as CNNs, RNNs, or DBNs. We further
expand our analysis by exploring a range of scholarly articles that employ
analogous time-series data [3, 4, 6, 7, 17, 18, 26, 28, 29, 30, 32, 33, 35,
46, 52, 58, 59, 60, 64].

###  4.1 Guide of Techniques Required for Research

In order for an undergraduate research to be successful in their research,
knowledge of some techniques is required. These techniques include an
understanding of a computer programming language such as Python and various
tools and packages.

####  4.1.1 Tools and Packages

Fluency in Python is extremely beneficial for replicating the works mentioned
in this paper effectively. Its readability, position as the de facto language
for ML, and extensive community support make it an excellent choice for
beginners in the EEG-ML field. NumPy and Pandas are useful ML libraries for
numerical operations and data manipulation, making them convenient for tasks
such as data preprocessing and exploration for EEG data. Overwhelmingly, the
aforementioned works employed the Python library Sci-kit Learn to train
classical ML algorithms such as Random Forests and SVMs. Deeper machine
learning algorithms such as CNNs and Transformers had split usage between the
PyTorch and Tensorflow/Keras Python libraries. Amazon Web Services (AWS), a
cloud computing solution, offers services such as SageMaker which offers a
complete ML development environment and EC2, an elastic virtual server
service, which enables the ability to launch and oversee virtual machine
instances to run ML code.

In summary, Python is quite mandatory for understanding EEG-ML source code;
Although not mandatory, familiarity and being comfirtable using Numpy, Pandas,
Sci-kit Learn, PyTorch or Tensorflow/Keras, are advantageous in further
enhancing the replication process. Finally, AWS or other cloud computing
services are nice tools to have especially when working with large EEG
datasets.

####  4.1.2 Recommended Courses

It is optional to enroll in a college course to learn Python. Many
universities such as Stanford, Cornell, and MIT provide free online courses
that are very efficient in their programming methods. Reading Python’s online
documentation is an alternative to gaining an proficiency of Python and its
syntax. Researchers lacking ML knowledge can also use online courses to gain
an understanding of the topic. Based on our own undergraduate research
experience, we recommend that undergraduate computer science researchers watch
Stanford CS 229 and CS 230 lectures online.

In addition, researchers should be comfortable utilizing NumPy, Pandas, Sci-
kit Learn, PyTorch, and Tensorflow/Keras libraries. We recommend that
researchers read each library’s documentation which outlines the capabilities
of each method in the libraries. These documentations can be found for free
online.

####  4.1.3 Recommended Papers

In addition to completing courses to understand the techniques and packages
needed to get started in BCI machine learning research, we recommended that
undergraduates read the papers outlined in table 5.

Recommended Paper Title | Paper
---|---
Deep learning-based electroencephalography analysis: a systematic review | [43]
Deep learning for electroencephalogram (EEG) classification tasks: a review | [5]
Fear of the CURE: A Beginner’s Guide to Overcoming Barriers in Creating |
a Course-Based Undergraduate Research Experience | [12]
Table 5: Recommended Papers to Start Research

###  4.2 Algorithms

####  4.2.1 Frameworks for Recommended Algorithms

CNNs are widely used in computer vision and image classifications. In [27],
they discuss a novel network that combines CNNs, SAE, and DNN to convert EEG
time series data into 2D images for good emotion classification. Once the EEG
data is transformed into 2D images, features extracted from the original EEG
data are sent to the CNN. The model includes many pooling layers and one
output layer. After pooling, data is subsampled into images with smaller
sizes. The weights and filters are learned through back-propagation. [27]

SVMs are a very useful supervised machine learning algorithm. It works by
separating the dataset into two sections. This sectioning of the data can
either be linear or nonlinear. Linear sectioning includes a discriminant
hyperplane, while nonlinear separation includes a kernel function. Though
computational complexity is low for this algorithm, its performance is heavily
dictated by the kernel function. [45]

Transformers are a trending algorithm for EEG classification. A transformer
network is a neural network that is based on a self-attention mechanism. The
network consists of encoders and decoders. Each encoder has two sub-layers, a
multi-head self-attention mechanism, and a position-wise connected forward
feeding network. There is a residual connection around the sub-layers, which
is followed by a normalization layer. [47]

####  4.2.2 Recommended Tasks

The tasks in Table 6 are recommended for undergraduate researchers to include
in their studies because they are extensively researched and have ML high
accuracy. The paper count column counts the number of papers that include the
use of transformers.

As shown in table 6, transformers can be used on a majority of Brain-Computer
Interface tasks, but are most commonly used for Motor Imagery, Emotion
Recognition, and Seizure Detection.

Task | Recommended Paper | Paper Count
---|---|---
Motor Imagery | [23] | 6
Seizure | [42] | 5
Emotion | [54] | 1
Table 6: Recommended Tasks

####  4.2.3 Recommended Algorithms

The algorithms in Table 7 are recommended. CNN is the most prominent; SVM is
low in computation cost, high in accuracy, and easy to understand.
Transformers are trending in BCI research, and have high accuracy and
parallelization.

Algorithm | Recommended Paper
---|---
CNN | [27]
SVM | [45]
Transformer | [47]
Table 7: Recommended Algorithms

####  4.2.4 Forms of Transformers

According to our finds, transformers are commonly accompanied with other
algorithms or are utilized in different forms. Table 8 outlines some of the
forms of transformers used from the papers found.

Algorithm | Dataset | Paper
---|---|---
CNN + Transformer | SEED | [13]
DNN + Transformer | TUEG | [22]
S3T | BCI Competition IV | [49]
Table 8: Combination of Algorithms with Transformers

For several emotion recognition tasks, transformers were combined with CNNs.
CNNs are fed the input DE features and extract complete information for multi-
frequency data with a depthwise convolution layer, which is then fed to a
transformer model. [13]

A combination of DNNs and Transformers was for a Motor Imagery task. DNNs were
used for pre-training, while transformers were used as the classification
algorithm. [22].

S3T is a Spatial-Temporal Tiny Transformer used for Motor Imagery tasks due to
its ability to capture spatial and temporal features. S3T requires an
attention mechanism to transform the data into a highly distinguishing
representation. The model consists of a spatial filter, spatial
transformation, temporal transformation, and a classifier. [49]

###  4.3 Datasets

####  4.3.1 Dataset Users

Each dataset utilizes different subjects with the goal to be classified for
different tasks. The goal of the BCI Competition datasets is to validate
signal processing and classification methods for Brain-Computer Interfaces
(BCIs). Each data set consists of single-trials of spontaneous brain activity,
one part labeled (calibration or training data) and another part unlabeled
(evaluation or test data), and a performance measure. The goal is to infer
labels (or their probabilities) for the evaluation data sets from calibration
data that maximize the performance measure for the true (but to the
competitors unknown) labels of the evaluation data.

The DEAP dataset is a multimodal dataset for the analysis of human affective
states. The EEG and peripheral physiological signals of 32 participants were
recorded as each watched 40 one-minute long excerpts of music videos.
Participants rated each video in terms of the levels of arousal, valence,
like/dislike, dominance and familiarity.

The SEED dataset contains subjects’ EEG signals when they were watching films
clips. The film clips are carefully selected so as to induce different types
of emotion, which are positive, negative, and neutral ones.

The STEW dataset consists of raw EEG data from 48 subjects who participated in
a multitasking workload experiment utilizing the SIMKAP multitasking test. The
subjects’ brain activity at rest was also recorded before the test and is
included as well.

####  4.3.2 Recommended Datasets

Table 9 overviews a recommended dataset based on the aforementioned
recommended tasks in Table 4.

Task | Recommended Dataset
---|---
Motor Imagery | EEGEyeNet
Emotion Recognition | DEAP
Seizure Detection | Bonn
Table 9: Dataset Recommendations

####  4.3.3 Dataset Breakdown For Transformers

From the papers found that utilize transformers, table 10 shows the breakdown
of the datasets that were used. The other category is a combination of
privately collected datasets or uncommon individual sets. DEAP and SEED are
available to the public.

Dataset | Paper Count
---|---
DEAP | 5
SEED | 2
Other | 8
Table 10: Dataset breakdown for Transformer Algorithm

###  4.4 Papers

####  4.4.1 Paper Recommendation for Undergraduate Researchers

We recommend 3 individual research papers in Table 11 and 3 review papers in
Table 12 to any new researcher to get acclimated with trends in BCI research.

Paper | Algorithm | Task
---|---|---
[47] | Transformer | Age/Gender
[14] | RF | Parkinson’s
[27] | CNN/AE | Emotion
Table 11: Individual Papers

There is limited available literature where EEG data has been used for gender
and age classification. The recommended paper for Age/Gender classification
gives a firm overview of the task as well as the algorithm. The paper
discusses how a transformer network was used for raw EEG data classification.
The utilization of Transformers in the classification aided in the achievement
of start-of-the-art accuracy for both gender and age. [47] Similarly, the
other two papers give a firm overview of their respective tasks, as well as a
clear insight into the framework and architecture of the algorithm.

Paper | Algorithm | Task
---|---|---
[44] | ML | BCI
[15] | DL | BCI
[16] | ML | Emotion
Table 12: Review Papers

Each of the papers highlighted in table 12 gives a strong overview of either
Machine Learning or Deep learning for their respective tasks. We recommend
that undergraduate researchers read these papers and get an even stronger
understanding of the BCI research landscape.

####  4.4.2 Paper Recommendation for Undergraduate BCI research using
Transformers

We recommended 3 individual research papers in Table 13. Each paper has a
different task and dataset. Each paper clearly explains its algorithms
framework, methodologies, and results.

Paper | Task | Dataset(s)
---|---|---
[51] | Motor Imagery | HysioNet
[31] | Emotion Recognition | DEAP/SEED
[8] | Seizure Detection | CHB-MIT
Table 13:  Recommended Transformer Papers

###  4.5 Challenges

One challenge encountered in utilizing ML algorithms for EEG task
classifications is their limited robustness. Often, these algorithms are
trained on data obtained from a single individual or a small group of
individuals. Consequently, when tested on data from a different individual who
was not part of the training set, ML algorithms tend to exhibit subpar
performance, requiring additional calibration and fine-tuning. To address this
issue, a potential solution lies in employing data augmentation techniques,
such as Generative Adversarial Networks (GANs). GANs can generate synthetic
EEG data, augmenting the existing dataset and enhancing the generality and
robustness of the models.

The practical utilization of brain waves for real-world task classifications
poses another challenge. Primarily, EEG data collection predominantly occurs
in controlled environments, which fails to capture the authentic real-world
conditions and factors that impact brain waves, including sensory stimuli and
varying levels of concentration. Consequently, ML algorithms often exhibit
under-performance when tested in real-world scenarios compared to their
performance on laboratory test data. It is imperative to undertake a
comprehensive evaluation and understanding of these factors during the design
phase to effectively integrate Brain-Computer Interfaces (BCIs) into real-
world applications within authentic environments.

###  4.6 Future Work

The utilization of transformers in classification models for BCI research has
demonstrated promising improvements in accuracy across a range of tasks,
rendering them applicable in the BCI and medical fields. Subsequent studies
can delve deeper into the analysis and exploration of transformers, expanding
upon the groundwork established by this paper, with the aim of enhancing their
performance by potentially mitigating computational requirements while further
improving accuracy. Similar avenues for improvement exist for other
algorithms, such as Recurrent Neural Networks (RNNs) and Convolutional Neural
Networks (CNNs).

In future research, it is imperative to investigate and identify algorithms
that exhibit the highest accuracy for specific tasks while also considering
their computational efficiency. This will provide clarity on which algorithms
are the most optimal in terms of both accuracy and computational requirements
for each task, thus enabling informed decisions in selecting the most suitable
algorithms for specific BCI applications.

##  5 Conclusion

In this paper, we have presented a comprehensive overview of essential tasks,
algorithms, and datasets that serve as foundational components for
undergraduate researchers embarking on EEG-based Brain-Computer Interface
(BCI) research using machine learning (ML).

Based on our extensive analysis, we recommend focusing on prominent tasks such
as Motor Imagery, Seizure Detection, and Emotion Classification. These tasks
offer a wealth of available datasets and exhibit high accuracy classification
rates. To assist researchers in getting started, we have identified specific
datasets that align with each task, providing valuable resources to facilitate
their investigations.

Furthermore, we suggest the incorporation of popular algorithms such as
Convolutional Neural Networks (CNNs), Support Vector Machines (SVMs), and
Transformers. These algorithms have consistently demonstrated remarkable
accuracy in EEG classification tasks, making them reliable choices for further
exploration. Specifically, we have highlighted the growing trend of
Transformers in EEG-based BCI research, providing an overview of the most
common task-dataset-algorithm combinations associated with this emerging
approach.

By following our recommendations, undergraduate researchers can establish a
solid foundation in the field, enabling them to confidently contribute to the
rapidly evolving landscape of BCI research. We believe that this systematic
overview and the suggested starting points will empower newcomers,
facilitating their exploration of novel techniques and inspiring their
contributions to advancements in BCI research.

As the field continues to progress, it is crucial to stay abreast of the
latest developments and embrace new opportunities for innovation. We hope that
this paper serves as a valuable resource, guiding researchers towards exciting
avenues and inspiring future discoveries in the dynamic realm of EEG-based
Brain-Computer Interfaces.

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