Limit for the Euler-genus distributions of ladder-like sequences of graphs

Abstract

In this paper, we demonstrate a method for calculating the production matrices for the Euler-genus polynomials of H-linear families of graphs. Particularly, for any ladder-like sequence of graphs, we give its explicit production matrix that leads to a recurrence relation. Based on this recurrence relation, we show that the Euler-genus distributions of any ladder-like sequence of graphs are asymptotic to a normal distribution. Since the genus polynomials of the ladder-like sequence of graphs have already been calculated by Chen et al.(J Algebr Combin 52:137–155, 2020), their crosscap-number polynomials are also known.

1 Introduction

In topology, surfaces are classified as either orientable surfaces $O_k(k\ge 0)$ with k handles or nonorientable surfaces $N_j(j\ge 1)$ with j crosscaps. The Euler-genus $\gamma ^E(S)$ of a surface S is defined by

$$\begin{aligned} \gamma ^E(S)= \left\{ \begin{aligned}&2k, \text { if } S=O_k, \\&j, \text { if }S=N_j. \end{aligned} \right. \end{aligned}$$

Let us denote by $S_i$ a surface with Euler-genus i for $i\ge 0$. The number of (distinct) cellular embeddings of a graph $G=(V(G),E(G))$ in the surface $S_i$ is denoted by $\varepsilon _i(G).$ By the Euler-genus distribution of a graph G, we mean the sequence

$$\begin{aligned} \varepsilon _0(G),\varepsilon _1(G),\varepsilon _2(G),\cdots . \end{aligned}$$

The Euler-genus polynomial of G is defined as the generating function $\mathcal {E}_G(x)=\sum _{i=0}^\infty \varepsilon _i(G)x^i.$

Determining the genus of graphs is one of the most important problems in topological graph theory, and the genus distributions or Euler-genus distributions of graphs have received considerable attention. For the genus distributions of graphs, we refer the reader to [6, 12,13,14, 17, 19, 21], etc. The research on the crosscap-number distribution was initiated by [1] according to the rank of Mohar’s algebraic invariant [15] and then developed by [3, 4], etc. Recently, Chen and Gross [5] introduced an Euler-genus approach to calculate the crosscap-number distributions. The objectives of most of the journal papers are to find recursions for the genus (Euler-genus) polynomials for specific families of graphs, closed formulas for those recursions, and proofs of log-concavity. Under some conditions, Zhang, Peng, and Chen [22] showed that the embedding distributions of H-linear families of graphs with spiders are asymptotic to normal distributions. It is also worth noting that Gross and Tucker [10] and White [20] provided the fundamentals of topological graph theory.

Our research mainly depends on the production matrices, which go back to Stahl [18]. In that paper, Stahl developed a technique called permutation-partition pairs to derive production matrices. Mohar [16] suggested that one can compute the production matrices using string operations directly. Independent of [16], using string operations, Gross et al. [11] gave the production matrices for the genus distributions of some families of graphs. Via production matrices, Chen, Gross, Mansour, and Tucker [7] obtained the genus polynomials of any ladder-like sequence of graphs.

The remainder of this paper is organized as follows. Section 2 introduces a method for finding the production matrices for the Euler-genus distributions of H-linear families of graphs. Section 3 applies this method to a ladder-like sequence of graphs and obtains its production matrix. In Sect.  4, we prove that the Euler-genus distributions of any ladder-like sequence of graphs are asymptotic to a normal distribution.

2 Production matrices

In this paper, we denote a root-vertex of a graph G by a single letter $a,b,\cdots ,$ a number $0,1,\cdots ,$ or a number $0, 1,\cdots$ with an overline. We represent a face of an embedding by a string of the root-vertices in the order in which they appear in a traversal of its boundary. According to the incidence of the face-boundary walks on the roots, the embeddings of G are partitioned into what we now call embedding types. Given an embedding type t,  we define its partial Euler-genus polynomial as

$$\begin{aligned} \sum _{i}a_ix^i, \end{aligned}$$

where $a_i$ is the number of type-t embeddings of G of Euler-genus i. If we order the embedding types, the partial Euler-genus distributions vector (or pEd-vector) for G is a column vector whose $r^{th}$ coordinate is the partial Euler-genus polynomial for the $r^{th}$ embedding type. These definitions are borrowed from Gross et al. [11], who also demonstrated some examples to explain these definitions. In the next subsection, we explore how a path-adding operation within a face or between faces affects the embedding types.

2.1 The rules for adding a path

Let G be a graph with designated root-vertices, and let uWv be a path whose endpoints u, v are roots of G but all other vertices of W are not in V(G). We adopt the following notational conventions for strings:

• The symbols P and Q stand for sequences of vertices of G and do not include any appearances of the designated u or v.

• The reverse string of Q is denoted by $Q^{-1}$.

• The strings P, Q, and W can be empty, and u may be equal to v. When W is an empty string, the operation of adding the path uWv simply involves adding an edge uv; hence, we write uv instead of uWv.

We emphasize here that all graphs considered in this paper are connected. Our rules for adding a path depend on the face-tracing algorithm. For the convenience of reading, we give a brief introduction to the face-tracing algorithm in [1].

(1)(Initiate). First, we choose a face corner $e_1ve_2$ that has not been traced and set $e_c=e_2,v_c=v,walktype=0.$

(2)(Advance one step). In the previous step, we arrived at $e_c,v_c$. Assume that the endpoints of $e_c$ are $\{v_c,v_{next}\}$ and that $\cdots e_le_ce_r\cdots$ is the rotation at $v_{next}$. Let $walktype=walktype\oplus \lambda (e_c)$ and $v_c=v_{next}$, where $\lambda$ is a mapping from E(G) to $\{0,1\}$ such that $\lambda (e)=0$ if e is untwist and $\lambda (e)=1$ otherwise. If $walktype=1,$ set $e_c=e_l$; otherwise, set $e_c=e_r$.

Repeat step 2 until the current face has been traced.

Fig. 1

Rule 1

Now, we introduce our rules.

Rule 1: When a path uWv is added between two different faces (uP), (vQ) of G, it follows that

$$\begin{aligned}{}[(uP),(vQ)]\!\!+\!\!uWv \!\! \rightarrow \!\! x^2(uPuWvQvW^{-1})\!\!+\!\!x^2(uPuWvQ^{-1}vW^{-1}). \end{aligned}$$

(1)

In view of $\beta (G)=|E(G)|-|V(G)|+1$, where $\beta (G)$ is the Betti number of G,  adding the path uWv between two different faces (uP), (vQ) of a connected graph G will result in a new graph $G+uWv$ with one more cotree edge than G, which is denoted by $\tilde{e}.$ By appropriate choice of the spanning tree, we can assume that the cotree edge $\tilde{e}$ lies in the path uWv. The blue circles in Fig. 1 denote the faces (uP) and (vQ). In the second step of the face-tracing algorithm, the value of $e_c$ ($e_c=e_l$ or $e_c=e_r$) depends not only on the value of $\lambda (e_c)$ but also on the history of face-tracing. Therefore, when we add a path uWv between two different faces (uP), (vQ) and trace the face from u in the counterclockwise direction, the resulting face has two possibilities depending on the value of $\lambda (\tilde{e})$ and the history of face-tracing, that is,

$$\begin{aligned} (uPuWvQvW^{-1}) \text { or }(uPuWvQ^{-1}vW^{-1}). \end{aligned}$$

(2)

If for the case $\lambda (\tilde{e})=0,$ the resulting face is one face in (2), then for the case $\lambda (\tilde{e})=1,$ the resulting face is the other face in (2). Combining these facts, we obtain Rule 1. For an embedding $\sigma :G\rightarrow S$, the famous Euler polyhedral equation states that

$$\begin{aligned} |V(G)|-|E(G)|+|F(\sigma )|=2-\gamma ^E(S), \end{aligned}$$

(3)

where $|F(\sigma )|$ stands for the number of faces of the embedding $\sigma$. Therefore, this way of adding a path requires two crosscaps or one handle more, which increases the Euler-genus by 2, with $x^2$ appearing in (1).

Now, we introduce our second rule.

Rule 2: When a path uWv is added within a face (uPvQ), it follows that

$$\begin{aligned}{}[(uPvQ)]+uWv\rightarrow (uPvW^{-1})(uWvQ)+x(uPvW^{-1}uQ^{-1}vW^{-1}). \end{aligned}$$

(4)

Fig. 2

Rule 2

With the same arguments as in Rule 1, we assume that there is one more cotree edge $\tilde{e}$, which lies in the path uWv. The blue circle in Fig. 2a denotes the face (uPvQ). After the path uWv is added, we trace the face from u in the counterclockwise direction. With the face-tracing algorithm, there are two possibilities depending on the value of $\lambda (\tilde{e}),$ that is, the faces $(uPvW^{-1})(uWvQ)$ in Fig. 2b or the face $(uPvW^{-1}uQ^{-1}vW^{-1})$ in Fig. 2c. We discuss in the following.

(1): The resulting faces are the faces in Fig. 2b for the case where $\lambda (\tilde{e})=0$. Then, for $\lambda (\tilde{e})=1$, the resulting face is the face in Fig. 2c. Considering the latter case ($\lambda (\tilde{e})=1$), by Euler polyhedral equation (3), this way of adding a path requires one crosscap more, which increases the Euler-genus by 1. Therefore, x appears in (4). By these discussions, we obtain Rule 2 in this situation.

(2): The resulting face is the face in Fig. 2c for the case where $\lambda (\tilde{e})=0$. Then, for $\lambda (\tilde{e})=1$, the resulting faces are the faces in Fig. 2b. Rule 2 also holds.

In the rest of this paper, we use $\text {Add}_{uWv}[(uP)(vQ)]$ and $\text {Add}_{uWv}[(uPvQ)]$ to denote the right sides of (1) and (4), respectively. Now, we give two remarks to explain our rules.

Remark 2.1

When a path uWv is added between two different faces (uP), (vQ), we do not know the resulting face just by the value of $\lambda (\tilde{e})$. However, $\lambda (\tilde{e})=0$ and $\lambda (\tilde{e})=1$ will result in two different faces, the total of which is given by (1). There are also similar arguments when we add a path uWv within a face (uPvQ).

Remark 2.2

Adding a path uWv that joins a face (uP) of an embedded graph to a face (vQ) of another embedded graph does not require an extra handle or crosscap. However, by taking this operation, the two faces will merge into a face. With similar discussions as for Rule 1, one of the following holds:

$$\begin{aligned}{}[(uP),(vQ)]+uWv\rightarrow (uPuWvQvW^{-1}), \end{aligned}$$

or

$$\begin{aligned}{}[(uP),(vQ)]+uWv\rightarrow (uPuWvQ^{-1}vW^{-1}). \end{aligned}$$

2.2 Suppressing and relabeling roots

The suppressing and relabeling operators were introduced in [11]. For a set of strings $\{i,j,\cdots \},$ the suppression operator $\text {Sup}_{i,j,\cdots }$ acts on an embedding type t and suppresses each appearance of $i,j,\cdots .$ For strings $i,i',j,j',\cdots$, the relabeling operator $\text {Relab}_{i\rightarrow i',j\rightarrow j',\cdots }$ acts on an embedding type t and replaces all i in t with $i'$, all j with $j',\cdots .$ We give two examples to show these two operators in action.

$$\begin{aligned}&\text {Sup}_{1,2,3}[(0)(0145)(2345)(123)(234)]=(0)(045)(45)(4).\\&\text {Relab}_{0\rightarrow 1,4\rightarrow 2}[(00)(3)(344)]=(11)(3)(322). \end{aligned}$$

2.3 Production matrices for H-linear families of graphs

Our definition of an H-linear family of graphs agrees with that given by Stahl [18]. Let H be any fixed graph. Roughly speaking, an H-linear family of graphs is a sequence of graphs $\{G_n:n=1,2,\cdots \}$ in which $G_n$ contains n copies of H that have been linked in a consistent way. The pEd-vector for $G_n$ is denoted by $V_n(x)$, and the production matrix M(x) is a matrix such that the following recursion holds:

$$\begin{aligned} V_n(x)=M(x)V_{n-1}(x),\quad n=2,3, \cdots . \end{aligned}$$

For a concrete H-linear family of graphs $\{G_n\}_{n=1}^\infty ,$ all possible embedding types can be listed. Then, applying the adding paths, suppressing embedding types and relabeling embedding types operators, we can obtain its production matrix.

We now give an example to show this procedure. For any $n\in \mathbb {N},$ $(J_n,0)$ is a cobblestone graph that has n 4-valent vertices and a 2-valent root-vertex 0. Figure 3 demonstrates this sequence of graphs $\{(J_n,0)\}_{n=1}^\infty$ with examples the $J_1,J_2,J_3$.

Fig. 3

Cobblestone graphs $(J_1,0)$(left),$(J_2,0)$(middle),and $(J_3,0)$(right)

Proposition 2.3

For $n\ge 1,$ the possible embedding types for embeddings of the cobblestone graph $(J_n,0)$ are (0)(0) and (00).

Example 2.4

Consider the sequence of cobblestone graphs $\{(J_n,0)\}_{n=1}^\infty .$ The production matrix for the Euler-genus distributions of $(J_{n},0)$ is

$$\begin{aligned} M (x)=\left[ \begin{matrix} 4&{}6 \\ 4x+4x^2&{}6x \end{matrix} \right] . \end{aligned}$$

(5)

Proof

To construct the graph $(J_{n},0)$ from $(J_{n-1},0)$, one needs the following operations:

• Add a path 010 from vertex 0 to vertex 0. By appropriate choice of the spanning tree, we can assume that the edge 01 in the path 010 is a cotree edge of the graph $J_{n-1}+010$.

• Suppress vertex 0.

• Relabel vertex 1 as vertex 0.

The composition of the above steps is denoted by $\text {Rec}_J$ for recursion. Now, we compute the effect of $\text {Rec}_J$ on the two embedding types (0)(0) and (00) for embeddings of $J_{n-1}.$

Case 1: The embedding type for embeddings of $J_{n-1}$ is (0)(0). Since we can add a path 010 between faces (0), (0) or within a face (0), by the operations in Rules 1 and 2 with P, Q as empty strings and $u=v=0,W=1$, we see that

$$\begin{aligned} \text {Add}_{010}[(0)(0)]= & {} 2x^2(001001)+2x^2(001001)\\&\quad +4(001)(01)(0)+4x(00101)(0). \end{aligned}$$

Then, we suppress the root 0, which yields

$$\begin{aligned}&\text {Sup}_{0}[(001001)]=(11),\text {Sup}_{0}[(001)(01)(0)]=(1)(1),\text {Sup}_{0}[(00101)(0)]=(1)(1). \end{aligned}$$

Combining the above operations, we conclude that

$$\begin{aligned}&\text {Sup}_{0}[\text {Add}_{010}[(0)(0)]]=4x^2(11)+4(1)(1)+4x(11). \end{aligned}$$

We apply the operator $\text {Relab}_{1\rightarrow 0}$ and obtain the following production:

$$\begin{aligned} \text {Rec}_J[(0)(0)]= \text {Relab}_{1\rightarrow 0} \big [\text {Sup}_{0}\big [\text {Add}_{010}[(0)(0)]]\big ] = (4x^2+4x)(00)+4(0)(0). \end{aligned}$$

(6)

Case 2: The embedding type for embeddings of $J_{n-1}$ is (00). There exist six different ways to add the path 010. By Rule 2, one sees that

$$\begin{aligned} \text {Add}_{010}[(00)]\!\!\!\!= & {} \!\!\!\! 4x(010001)+2x(001001)+2(001)(001)+4(01)(0001). \end{aligned}$$

Then, we suppress the root 0 and arrive at

$$\begin{aligned}&\text {Sup}_{0}[(010001)]=(11), \text {Sup}_{0}[(001001)]=(11),\nonumber \\&\text {Sup}_{0}[(001)(001)]=(1)(1), \text {Sup}_{0}[(01)(0001)]=(1)(1). \end{aligned}$$

(7)

Combining the above operations, we conclude that

$$\begin{aligned}&\text {Sup}_{0}[\text {Add}_{010}[(00)]]=6x(11)+6(1)(1). \end{aligned}$$

Applying the operator $\text {Relab}_{1\rightarrow 0}$, we obtain the following production:

$$\begin{aligned} \text {Rec}_J[(00)]= \text {Relab}_{1\rightarrow 0} \big [\text {Sup}_{0}[\text {Add}_{010}[(00)]]\big ] =6x(00)+6(0)(0). \end{aligned}$$

(8)

Recording the coefficients of (0)(0), (00) in productions (6)(8) as columns of the production matrix M(x) for $\text {Rec}_J$ gives (5). Accordingly, by direct calculations, the pEd-vectors for $J_1,J_2$ are given by

$$\begin{aligned} V_{J_1}=\left[ \begin{matrix} 4+6x \\ 4x+10x^2 \end{matrix} \right] , V_{J_2}=\left[ \begin{matrix} 16+48x+60x^2 \\ 16x+64x^2+84x^3 \end{matrix} \right] . \end{aligned}$$

3 Euler-genus polynomials of a ladder-like sequence of graphs

In this section, we consider Euler-genus polynomials of a ladder-like sequence of graphs. We adopt the same definition of a ladder-like sequence of graphs as that in [7]. Let (H, u, v) be a graph with 1-valent root-vertices u and v, and let the shadow part of H be any connected graph. Figure 4 demonstrates the ladder-like sequence of graphs $\{(L_{n}^H, u_{n}, v_{n})\}_{n=1}^\infty$. Note that for $n\ge 2,$ the root-vertices $u_{n}, v_{n}$ for $\{(L_{n}^H, u_{n}, v_{n})\}$ are both 2-valent. However, the initial graph $L_1^{H}$ is isomorphic to (H, u, v), in which the root-vertices u, v are 1-valent. To compensate for this shortcoming, Chen, Gross, Mansour, and Tucker [7] defined the extended super-rung $H^+$, which is a graph obtained by attaching an edge at the root-vertex u of H and another edge at the root-vertex v of H. Then, the root-vertices u, v of $H^+$ are 2-valent.

Fig. 4

Graph H (left), and ladder-like graph $L^H_n$ (right)

From now on, the root-vertices u and v of (H, u, v) are denoted by 0 and 1, respectively, that is, (H, 0, 1). Similar arguments apply to $(L_{n}^H, u_{n},v_{n})$ and $(H^+,u,v)$.

Proposition 3.1

The possible embedding types for embeddings of the graph (H, 0, 1) are (0)(1) and (01).

Proposition 3.2

The possible embedding types for embeddings of the ladder-like graph $(L_{n}^H,0,1)$ are

$$(00)(11),(01)(01),(0011),(0101).$$

Proof

Since the roots 0 and 1 are 2-valent, there are 10 possible embedding types. According to the definition of the ladder-like graph $L_n^{H}$, the roots 0 and 1 form a cutset. Thus, a possible embedding type for the embedding of $L_n^{H}$ has no 1-cycles, and the following six embedding types are impossible:

$$ (0)(0)(1)(1),(0)(0)(11),(0)(1)(01),(1)(1)(00),(0)(011),(1)(001).$$

The proof is completed. $\square$

Theorem 3.1

Let (H, 0, 1) be any connected graph with two 1-valent root-vertices 0 and 1. The partial Euler-genus polynomials for H of embedding types (0)(1) and (01) are denoted by p(x) and q(x), respectively. Then, (i) the production matrix for the Euler-genus distributions of the ladder-like sequence of graphs $\{L_n^H\}_{n=1}^\infty$ is

$$\begin{aligned} M_L^H(x)=p(x)\left[ \begin{matrix} 8x^2&{}4x^2 &{}0&{} 0 \\ 0&{} 0&{} 0 &{}0 \\ 0 &{}4x^2&{} 8x^2&{} 8x^2 \\ 0&{} 0&{} 0&{} 0 \end{matrix} \right] +q(x)\left[ \begin{matrix} 0&{}0 &{}0&{}0 \\ 0&{} 2&{} 4&{} 4 \\ 8x^2 &{}4x^2&{} 0&{}0 \\ 0 &{} 2x&{} 4x&{} 4x \end{matrix} \right] ; \end{aligned}$$

and (ii) for $n\ge 2$, the pEd-vector for $L_n^H$ is $M_L^H(x)^{n-1}X_1$, where

$$X_1=[p(x)/4,0,q(x)/4,0]^T$$

.

Proof

To avoid confusion, we write the roots of $L_{n-1}^H$ as $\bar{0}$ and $\bar{1}$. The embedding of the graph $(L_{n-1}^H, \bar{0},\bar{1})$ must be one of the four embedding types $(\bar{0}\bar{0})(\bar{1}\bar{1}),$ $(\bar{0}\bar{1})(\bar{0}\bar{1}),$ $(\bar{0}\bar{0}\bar{1}\bar{1}),$ $(\bar{0}\bar{1}\bar{0}\bar{1})$. We carry out the following procedures to obtain the pEd-vector of $L_n^H$ from the pEd-vector of $L_{n-1}^H.$ Figure 5 presents an outline of our procedures.

Fig. 5

Adding edges $\bar{0}0$ and $\bar{1}1$

Step 1: Add the edge $\bar{0}0$, and then suppress the root $\bar{0}$. Through this step, a face of the embedded graph $L_{n-1}^H$ merges with a face of the embedded graph H and forms a new face.

Note that for (H, 0, 1), there are two embedding types (0)(1) and (01). Consider the embedding type $(\bar{0}\bar{0})(\bar{1}\bar{1})$ for any given embedding of $L_{n-1}^H$ and the embedding type (0)(1) for any given embedding of H. Adding the edge $\bar{0}0$ and suppressing the root $\bar{0}$, by Remark 2.2, we obtain the following production:

$$\begin{aligned} \quad \quad \text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{0})(\bar{1}\bar{1}),(0)(1)]]= \text {Sup}_{\bar{0}}[ 2(\bar{0}\bar{0}\bar{0}00)(\bar{1}\bar{1})(1)]=2(00)(\bar{1}\bar{1})(1).\nonumber \\ \end{aligned}$$

(9)

Similarly, we obtain the following seven productions:

$$\begin{aligned} \begin{aligned}&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{1})(\bar{0}\bar{1}),(0)(1)]]= 2(00\bar{1})(\bar{1})(1),\\&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{0}\bar{1}\bar{1}),(0)(1)]]= 2(00\bar{1}\bar{1})(1),\\&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{1}\bar{0}\bar{1}),(0)(1)]]= 2(00\bar{1}\bar{1})(1),\\&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{0})(\bar{1}\bar{1}),(01)]]= 2(010)(\bar{1}\bar{1}),\\&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{1})(\bar{0}\bar{1}),(01)]]= 2(010\bar{1})(\bar{1}),\\&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{0}\bar{1}\bar{1}),(01)]]= 2(010\bar{1}\bar{1}),\\&\text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{1}\bar{0}\bar{1}),(01)]]= 2(010\bar{1}\bar{1}). \end{aligned} \end{aligned}$$

(10)

Step 2: We add edge $\bar{1}1$ and suppress the root $\bar{1}$. Let us explain the effect of this step on the embedding types in detail.

First, we consider the embedding type $(\bar{0}\bar{0})(\bar{1}\bar{1})$ for any given embedding of $L_{n-1}^H$ and the embedding type (0)(1) for any given embedding of H. Recall that in Step 1, we obtained

$$\begin{aligned} \text {Sup}_{\bar{0}}[\text {Add}_{\bar{0}0}[(\bar{0}\bar{0})(\bar{1}\bar{1}),(0)(1)]]=2(00)(\bar{1}\bar{1})(1). \end{aligned}$$

(11)

Then, for this intermediate embedding type $(00)(\bar{1}\bar{1})(1)$, we proceed to take the operations of adding the edge $\bar{1}1$ and suppressing the root $\bar{1}.$ Thanks to Rule 1, we have the following production:

$$\begin{aligned} \text {Sup}_{\bar{1}} [\text {Add}_{\bar{1}1}[(00)(\bar{1}\bar{1})(1)]]= \text {Sup}_{\bar{1}} [4x^2(\bar{1}\bar{1}\bar{1}11)(00)]=4x^2(00)(11). \end{aligned}$$

(12)

Since there are p(x) embeddings of H of embedding type (0)(1), combining (11) and (12), we arrive at

$$\begin{aligned} (\bar{0}\bar{0})(\bar{1}\bar{1})+p(x)(0)(1)\xrightarrow []{~~\text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0} ~~} 2p(x)(00)(\bar{1}\bar{1})(1) \xrightarrow []{~~\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}~~} 8p(x) x^2(00)(11). \end{aligned}$$

With similar arguments to those given in the above paragraph, by (10) and Rules 1 and 2, we also obtain the following seven productions:

$$\begin{aligned} (\bar{0}\bar{1})(\bar{0}\bar{1}) \!\!+\!\!p(x)(0)(1)&\!\! \xrightarrow []{\text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0}} \!\! 2p(x)(00\bar{1})(\bar{1})(1) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}} \!\! 4p(x)x^2((00)(11)+(0011)),\\ (\bar{0}\bar{0}\bar{1}\bar{1})\!\!+\!\!p(x)(0)(1)&\!\!\xrightarrow []{ \text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0} }\!\! 2p(x)(00\bar{1}\bar{1})(1) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}}\!\! 8p(x)x^2(0011),\\ (\bar{0}\bar{1}\bar{0}\bar{1})\!\!+\!\!p(x)(0)(1)&\!\! \xrightarrow []{ \text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0}} \!\! 2p(x)(00\bar{1}\bar{1})(1) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}}\!\! 8p(x)x^2(0011),\\ (\bar{0}\bar{0})(\bar{1}\bar{1})\!\!+\!\!q(x)(01)&\!\! \xrightarrow []{\text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0} } \!\! 2q(x)(010)(\bar{1}\bar{1}) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}} \!\! 8q(x)x^2(0011),\\ (\bar{0}\bar{1})(\bar{0}\bar{1})\!\!+\!\!q(x)(01)&\!\! \xrightarrow []{ \text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0} }\\&2q(x)(010\bar{1})(\bar{1}) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}}\!\! 2q(x)\big (2x^2(0011)\!+\!(01)(01)\!+\!x(0101)\big ),\\ (\bar{0}\bar{0}\bar{1}\bar{1})\!\!+\!\!q(x)(01)&\!\! \xrightarrow []{\text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0} } \!\! 2q(x)(010\bar{1}\bar{1}) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}}\!\! 4q(x)(01)(01)+4xq(x)(0101),\\ (\bar{0}\bar{1}\bar{0}\bar{1})\!\!+\!\!q(x)(01)&\!\! \xrightarrow []{ \text {Sup}_{\bar{0}}\circ \text {Add}_{\bar{0}0}} \!\! 2q(x)(010\bar{1}\bar{1}) \!\! \xrightarrow []{\text {Sup}_{\bar{1}}\circ \text {Add}_{\bar{1}1}}\!\! 4q(x)(01)(01)+4xq(x)(0101). \end{aligned}$$

As there are q(x) embeddings of H of embedding type (01), q(x) appears in the last four productions.

By these eight productions in Step 2, the statement in (i) is proved.

Now, we are in a position to prove (ii). Since the partial Euler-genus polynomials for H of embedding types (0)(1) and (01) are p(x) and q(x), respectively, the partial Euler-genus polynomials of the graph $(H^+,0,1)$ are p(x), 0, q(x),  and 0 for embedding types (00)(11), (01)(01),  (0011),  and (0101), respectively. That is,

$$[p(x),0 ,q(x),0]^T$$

is the pEd-vector of $H^+$.

For the graph $L_n^H$ in Fig. 4, we attach an edge at each red vertex and call the resulting graph the extension of $L_n^H$. Note that the two added edges cannot be cotree edges of the extension of $L_n^H$ and that the extension of $L_n^H$ has four embeddings for each embedding of $L_n^H.$ The statement in (ii) of Theorem 3.1 is thus proved. $\square$

Remark 3.3

Since H is a connected graph, $q(x)=0$ is impossible.

Example 3.4

Let the shadow part of H be a 2-cycle. The partial Euler-genus polynomials are

$$\begin{aligned} \mathcal {E}_H^{(0)(1)}(x)=2, \,\,\,\, \mathcal {E}_H^{(01)}(x)=2+4x. \end{aligned}$$

The pEd-vector of H is

$$\begin{aligned} \left[ \begin{matrix} 2 \\ 2+4x \end{matrix} \right] . \end{aligned}$$

Therefore, $p(x)=2$ and $q(x)=2+4x$.

Owing to Proposition 3.2, there are four embedding types $(00)(11),(01)(01),$ $(0011),$ (0101) for the embeddings of $L_n^{H}$. Let

$$X_1=[2/4,0,(2+4x)/4,0]^T.$$

Due to Theorem 3.1, the production matrix $M_L^H(x)$ is

$$\begin{aligned} \left[ \begin{matrix} 16x^2&{}8x^2&{}0&{}0 \\ 0&{}8x+4&{}16x+8&{}16x+8 \\ 32x^3+16x^2&{}16x^3+16x^2&{}16x^2&{}16x^2 \\ 0&{}8x^2+4x&{}16x^2+8x&{}16x^2+8x \end{matrix} \right] , \end{aligned}$$

and the pEd-vector of $L_n^H$ is $M_L^H(x)^{n-1}X_1$.

Theorem 3.2

The following recurrence relation holds for $\mathcal {E}_{L_n^H},n\ge 3$:

$$\begin{aligned} \begin{aligned}&\mathcal {E}_{L_n^H}(x)=\big (16x^2 p(x)+2q(x)+4xq(x)\big )\mathcal {E}_{L_{n-1}^H}(x) \\&-\big (64 x^4 p^2(x)+16 x^2 p(x) q(x)+32 x^3 p(x) q(x) - 16 x^2 q^2(x)\big )\mathcal {E}_{L_{n-2}^H}(x). \end{aligned} \end{aligned}$$

(13)

Furthermore, we have the following explicit expression for

$\mathcal {E}_{L_n^H}(x)$:

$$\begin{aligned} \mathcal {E}_{L_n^H}(x)= & {} C_1(x)\left( 8 x^2 p(x)+\sqrt{20 x^2 q(x)^2+4 x q(x)^2+q(x)^2}+2 x q(x)+q(x)\right) ^n\\&+C_2(x) \left( 8 x^2 p(x)-\sqrt{20 x^2 q(x)^2+4 x q(x)^2+q(x)^2}+2 x q(x)+q(x)\right) ^n, \end{aligned}$$

where

$$\begin{aligned}&C_1(x)\!\! =\!\! \frac{ \Big (8 x^2 p(x)+(-\sqrt{20 x^2+4 x+1}+2 x+1) q(x)\Big )}{128 x^2 \sqrt{20 x^2+4 x+1} \big ((2 x+1) p(x) q(x)+4 x^2 p(x)^2-q(x)^2\big )}\cdot \\&\quad \quad \quad \Big ((8 x^2+\sqrt{20 x^2+4 x+1}-2 x-1) p(x)+(\sqrt{20 x^2+4 x+1}+2 x+3) q(x)\Big ),\\&C_2(x)\!\!=\!\!-\frac{\left( -8 x^2+\sqrt{20 x^2+4 x+1}+2 x+1\right) p(x)+\left( \sqrt{20 x^2+4 x+1}-2 x-3\right) q(x)}{8 \sqrt{20 x^2+4 x+1} \big ((\sqrt{20 x^2+4 x+1}-2 x-1) q(x)-8 x^2 p(x)\big )}. \end{aligned}$$

Proof

By Theorem 3.1, the characteristic polynomial of the production matrix $M_L^H(x)$ is

$$\begin{aligned} F(x,\lambda )= & {} \lambda ^4+\big (-16x^2 p(x)-2q(x)-4xq(x)\big )\lambda ^3\\&+\big (64 x^4 p^2(x)+16 x^2 p(x) q(x)+32 x^3 p(x) q(x) - 16 x^2 q^2(x)\big )\lambda ^2. \end{aligned}$$

Then, following the lines in [8, Page 6-Page 7] and noting

$$\begin{aligned} \mathcal {E}_{L_n^H}(x)=(1,1,\cdots ,1) M_L^H(x)^{n-1}X_1, \end{aligned}$$

we obtain (13).

With the help of a computer, for any fixed x, the solutions to equation $F(x,\lambda )=0$ are given by

$$\begin{aligned}&\lambda _1(x)=8 x^2 p(x)+\sqrt{20 x^2 q(x)^2+4 x q(x)^2+q(x)^2}+2 x q(x)+q(x), \\&\lambda _2(x)= 8 x^2 p(x)-\sqrt{20 x^2 q(x)^2+4 x q(x)^2+q(x)^2}+2 x q(x)+q(x), \\&\lambda _3(x)=\lambda _4(x)=0. \end{aligned}$$

Then, the general solution to (13) is

$$\begin{aligned} \mathcal {E}_{L_n^H}(x)= & {} C_1(x) \lambda _1^n(x)+C_2(x)\lambda _2^n(x). \end{aligned}$$

Combining Theorem 3.1 (ii) and the above equality, we can derive the expressions of $C_1(x),C_2(x)$ with the help of a computer. The proof is completed. $\square$

Noting that the genus polynomials of a ladder-like sequence of graphs have already been calculated by [7, 11], by our results, their crosscap-number polynomials are also known.

4 The limit of Euler-genus distributions for ladder-like graphs

Before we state our main results, we give some notations and provide a simple version of [22, Theorem 3.5], which is a main tool in this section.

Let $\mathcal {G}=\{G_n\}_{n=1}^\infty$ be a H-linear family of graphs. For Euler-genus distributions, we denote the production matrix of $\mathcal {G}$ by M(x). Assume that the maximum eigenvalue of M(x) is $\lambda _1(x)$, and set

$$\begin{aligned}&e=\frac{\lambda _1'(1)}{\lambda _1(1)}, \quad v=\frac{-\big (\lambda _1'(1)\big )^2 +\lambda _1(1)\cdot \lambda _1^{''}(1) +\lambda _1(1)\cdot \lambda _1'(1) }{\lambda _1(1)^2}. \end{aligned}$$

(14)

Zhang, Peng, and Chen [22] proved the following proposition.

Proposition 4.1

Consider the H-linear family of graphs $\mathcal {G}=\{G_n\}_{n=1}^\infty$. If $v>0$ and, for the matrix M(1), the algebraic multiplicity of $\lambda _1(1)$ is one, then the Euler-genus distributions of $G_n$ are asymptotically the normal distributions with mean $e\cdot n$ and variance $v\cdot n$ when n approaches infinity. That is,

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{x\in \mathbb {R}}\left| \frac{1}{\mathcal {E}_{G_n}(1)}\sum _{\tiny {0\le i\le x \sqrt{ v\cdot n}+e\cdot n]}}\varepsilon _i(G_n)-\int _{-\infty }^x \frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2}u^2}\mathrm {d} u\right| =0. \end{aligned}$$

Adopting the same notations as in Theorem 3.1, let (H, 0, 1) be any connected graph with two 1-valent root-vertices, and denote the partial Euler-genus polynomials for H of embedding types (0)(1) and (01) by p(x) and q(x), respectively.

The aim of this section is to prove the following theorem.

Theorem 4.1

The Euler-genus distributions of the ladder-like sequence of graphs $L_n^H$ are asymptotic to the normal distributions with mean $e\cdot n$ and variance $e\cdot n$, where

$$\begin{aligned} \quad e= & {} \,\frac{5 \left( p'(1)+q'(1)\right) +10 p(1)+4 q(1)}{5 (p(1)+q(1))},\\ \nonumber \quad v= & {} \,\frac{1}{64(p(1)+q(1))}\Big [ 22 q(1)^2-125 \big (p'(1)+q'(1)\big )^2 \end{aligned}$$

(15)

$$\begin{aligned}&\quad +25 q(1) \big (5 p''(1)+5q''(1)+5q'(1)+17 p'(1)\big )\\ \nonumber&\quad + p(1) \big (125 p''(1)+125 p'(1)+125 q''(1)-175 q'(1)+202 q(1)\big ) \Big ]. \end{aligned}$$

(16)

Proof

Our strategy is to verify these conditions in Proposition 4.1.

First, by Theorem 3.1, the production matrix for Euler-genus distributions of the ladder-like sequence of graphs $L_1^H,L_2^H,L_3^H,\cdots$ is

$$\begin{aligned} M_L^H(x)=\left[ \begin{matrix} 8 x^2 p(x) &{} 4 x^2 p(x) &{} 0 &{} 0 \\ 0 &{} 2 q(x) &{} 4 q(x) &{} 4 q(x) \\ 8 x^2 q(x) &{} 4 p(x) x^2+4 q(x) x^2 &{} 8 x^2 p(x) &{} 8 x^2 p(x) \\ 0 &{} 2 x q(x) &{} 4 x q(x) &{} 4 xq(x) \\ \end{matrix} \right] . \end{aligned}$$

By direct calculation, the eigenvalues of the matrix $M_L^H(1)$ are given by

$$\begin{aligned} \lambda _1=8(p(1)+q(1)),~\lambda _2=8p(1)-2q(1),\lambda _3=0,\lambda _4=0. \end{aligned}$$

In view of Remark 3.3, we have $p(1)\ge 0$ and $q(1)>0$, which yield

$$\begin{aligned} \lambda _1>|\lambda _2|, \quad \lambda _1>|\lambda _3|, \quad \lambda _1>|\lambda _4|. \end{aligned}$$

(17)

Now, we can show that $v>0.$ With the help of a computer, the maximum eigenvalue of $M_L^H(x)$ is

$$\begin{aligned} \lambda _1(x)= 8 x^2 p(x)+\sqrt{20 x^2 q(x)^2+4 x q(x)^2+q(x)^2}+2 x q(x)+q(x), \end{aligned}$$

and the values of

$$\begin{aligned}&e= \frac{\lambda _1'(1)}{\lambda _1(1)}, \quad v=\frac{-\big (\lambda _1'(1)\big )^2 +\lambda _1(1)\cdot \lambda _1^{''}(1) +\lambda _1(1)\cdot \lambda _1'(1) }{\lambda _1(1)^2} \end{aligned}$$

are given by (15) and (16), respectively. Assume that

$$\begin{aligned} p(x)=\sum _{i=0}^\infty a_ix^i,\quad q(x)=\sum _{j=0}^\infty b_jx^j. \end{aligned}$$

By the Cauchy-Schwarz inequality, one sees that $p'(1)^2\le (p''(1)+p'(1))\cdot p(1)$ and $q'(1)^2\le (q''(1)+q'(1))\cdot q(1)$. Combining these inequalities with (16), to show $v>0$, it suffices to show that

$$\begin{aligned}&250p'(1)q'(1)+175p(1)q'(1)\\&< 22 q(1)^2 + q(1) \Big (125 p''(1)+425 p'(1)\Big ) +p(1) \Big (125 q''(1)+202 q(1)\Big ), \end{aligned}$$

which is equivalent to

$$\begin{aligned}&\sum _{i,j}\Big [250ija_ib_j+175a_ijb_j\Big ] <2 q(1)^2 \\&\quad + \sum _{i,j}\Big [125(i^2-i)+425i+125(j^2-j)+202\Big ]a_ib_j. \end{aligned}$$

The above inequality is due to

$$\begin{aligned}&2 q(1)^2+ \sum _{i,j}\Big [125(i-j)^2+300(i-j)+202\Big ]a_ib_j\\&= 2 q(1)^2+ \sum _{i,j}\Big (125\big (i-j+\frac{6}{5}\big )^2+22\Big )a_ib_j>0, \end{aligned}$$

where we have used Remark 3.3 in the last inequality. We complete the verification of $v>0.$

By (17) and $v>0,$ applying Proposition 4.1 to the ladder-like sequence of graphs $\{L_n^H\}_{n=1}^\infty ,$ we complete our proof. $\square$

Finally, we demonstrate some research problems.

Question 4.2

For a star-ladder $S\!L_{n_1,n_2,\cdots ,n_k}$ with signature $\{n_1,n_2,\cdots ,n_k\},$ Chen, Gross, and Mansour [2] derived a formula for its genus distribution. Can we prove that the genus distributions of $S\!L_{n_1,n_2,\cdots ,n_k}$ are asymptotic to normal distributions when $N=n_1+n_2+\cdots +n_k$ approaches infinity.

Question 4.3

Consider the ladder-like sequence of graphs $L_n^H$. With consideration of the probability, the Euler-genus distribution of $L_n^H$ is a 1-dimensional distribution, and the partial Euler-genus distribution of $L_n^H$ is a 4-dimensional distribution. Can we prove that the limit of the partial Euler-genus distribution of $L_n^H$ is a 4-dimensional normal distribution?

Question 4.4

It has been conjectured in [12] that the genus polynomial of every graph is log-concave. Can we use Theorem 4.1 to prove that the Euler-genus polynomial of $L_n^H$ is log-concave when n is large enough?

Question 4.5

Proposition 4.1 requires that $v>0$ and that the algebraic multiplicity of $\lambda _1(1)$ be one. Can we consider the case where $v=0$ or the algebraic multiplicity of $\lambda _1(1)$ is larger than one and obtain similar results?

Question 4.6

Consider the H-linear family of graphs $\mathcal {G}=\{G_n\}_{n=1}^\infty$. Let

$$\begin{aligned} A_n= \sup _{x\in \mathbb {R}}\left| \frac{1}{\mathcal {E}_{G_n}(1)}\sum _{\tiny {0\le i\le x \sqrt{ v\cdot n}+e\cdot n]}}\varepsilon _i(G_n)-\int _{-\infty }^x \frac{1}{\sqrt{2\pi }}e^{-\frac{1}{2}u^2}\mathrm {d} u\right| , \end{aligned}$$

where the constants e, v appear in (14). Under some conditions, Proposition 4.1 states that $A_n$ tends to 0 when n tends to infinity. Can we give some estimates of the rate of convergence?