Spanning Trees of Bounded Degree, Connectivity, Toughness, and the Spectrum of a Graph

Abstract

Recently, Cioabǎ and Gu obtained a relationship between the spectrum of a regular graph and the existence of spanning trees of bounded degree, generalized connectivity and toughness, respectively. In this paper, motivated by the idea of Cioabǎ and Gu, we determine a connection between the (signless Laplacian and Laplacian) eigenvalues of a graph and its structural properties involving the existence of spanning trees with bounded degrees and generalized connectivity, respectively. We also present a connection between the (signless Laplacian and Laplacian) eigenvalues and toughness of a bipartite graph, respectively. Finally, we obtain a lower bound of toughness in a graph in terms of edge connectivity $\kappa '$ and maximum degree $\varDelta$.

1 Introduction

Throughout this paper, we only consider finite undirected simple graphs. We follow the terminology and notation of Bondy and Murty [3] or Brouwer and Haemers [6] for graphs, unless otherwise defined. Let G be a connected graph on n vertices with vertex set $\{v_{1},v_{2},\ldots ,v_{n}\}$ and edge set $\{e_{1},e_{2},\ldots ,e_{m}\}$. Let $d_{i}=d_{i}(G)$ denote the degree of a vertex $v_{i}\in V(G)$. Let $\delta =\delta (G)$ and $\varDelta =\varDelta (G)$ be the minimum degree and the maximum degree of a graph G, respectively. If $d_{i}=d$ for each vertex $v_{i}\in V(G)$, then G is called a d-regular graph.

The adjacency matrix of G is an $n\times n$ matrix $A(G)=(a_{ij})$, where $a_{ij}$ is the number of edges joining $v_{i}$ and $v_{j}$ for $1\le i,j \le n$. If G is simple, then A(G) is a symmetric (0,1)-matrix. The eigenvalues of A(G) are the eigenvalues of G. We use $\lambda _{i}=\lambda _{i}(G)$ to denote the ith largest eigenvalue of G. Therefore, we assume $\lambda _{1} \ge \lambda _{2} \ge \cdots \ge \lambda _{n}$. Let $\lambda =\max \{|\lambda _{2}|,|\lambda _{n}|\}.$ Let $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ be the Laplacian matrix and the signless Laplacian matrix of G, respectively, where $D(G)=diag(d_{1}, d_{2}, \ldots , d_{n})$ is the diagonal matrix of vertex degree of G. We use $\mu _{i}=\mu _{i}(G)$ and $q_{i}=q_{i}(G)$ to denote the ith largest eigenvalue of L(G) and Q(G), respectively.

For a graph G, let $\kappa (G), \kappa '(G)$ and c(G) denote the vertex connectivity, the edge connectivity and the number of components of G, respectively. For an integer $k \ge 2$, a k-tree is a tree with the maximum degree at most k. For an integer $l \ge 2$, Chartrand et al. [7] defined the l-connectivity $\kappa _{l}(G)$ of G to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Thus, $\kappa _{l}(G) = 0$ if and only if $c(G) \ge l$ or $\mid V(G)\mid \le l-1$. If $\kappa _{l}(G)\ge k$ for $k\ge 1$, then G is called (k, l)-connected. The toughness t(G) of a connected graph G is defined as $t(G) = min\{|S|\times (c(G-S))^{-1}\}$, where the minimum is taken over all proper subsets $S \subseteq V(G)$ such that $c(G-S) > 1$. If $t(G) \ge t$, then G is called t-tough. The girth of G is defined as

$$\begin{aligned} g=g(G)= \left\{ \begin{array}{ll} \min \{|E(C)|: C \text{ is } \text{ a } \text{ cycle } \text{ of } G\}, &{}\quad \text{ if } G \text{ is } \text{ not } \text{ acyclic },\\ \infty , &{}\quad \text{ if } G \text{ is } \text{ acyclic }.\\ \end{array}\right. \end{aligned}$$

Wong [24] gave a relationship between the eigenvalues and the existence of spanning k-trees in a regular graph.

Theorem 1.1

[24] Let G be a connected d-regular graph and let $k \ge 3$. If

$$\begin{aligned} \lambda _{4}<d-\frac{d}{(k-2)(d+1)}, \end{aligned}$$

then G has a spanning k-tree.

Recently, Cioabǎ and Gu [8] further improved the above result.

Theorem 1.2

[8] Let G be a connected d-regular graph with edge connectivity $\kappa '$, and let $k \ge 3$. Let $l=d-(k-2)\kappa '.$ Each of the following statements holds.

• (i) If $l \le 0,$ then G has a spanning k-tree.

• (ii) If $l>0$ and $\lambda _{\lceil \frac{3d}{l}\rceil }<d-\frac{d}{(k-2)(d+1)}$, then G has a spanning k-tree.

The following Theorem 1.3 presents a relationship between the eigenvalues and the existence of spanning k-trees in a graph. This result generalizes Theorem 1.2 from a regular graph to a graph.

Theorem 1.3

Let G be a connected graph with maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '$, and let $k \ge 3$. Let $l=\varDelta -(k-2)\kappa '$. Each of the following statements holds.

• (i) If $l \le 0,$ then G has a spanning k-tree.

• (ii) If $l > 0$, $\varDelta \le (k-2)\delta$ and $\lambda _{\lceil \frac{3\varDelta }{l}\rceil }\le \delta -\frac{\varDelta }{(k-2)(\delta +1)},$ then G has a spanning k-tree.

Cioabǎ and Gu [8] gave a relationship between the eigenvalues and generalized connectivity in a regular graph.

Theorem 1.4

[8] Let l, k be integers such that $l \ge k \ge 2.$ For any connected d-regular graph G with $\mid V(G)\mid \ge k+l-1$, $d \ge 3$ and edge connectivity $\kappa '$, if $\kappa ' = d,$ or if $\kappa ' < d$ and

$$\begin{aligned} \lambda _{\lceil \frac{(l-k+1)d}{d-\kappa '}\rceil }(G)< \left\{ \begin{array}{ll} \frac{d-2+\sqrt{d^{2}+12}}{2}, &{}\quad \text{ if } d \text{ is } \text{ even },\\ \frac{d-2+\sqrt{d^{2}+8}}{2}, &{}\quad \text{ if } d \text{ is } \text{ odd },\\ \end{array}\right. \end{aligned}$$

then $\kappa _{l}(G)\ge k.$

We mainly generalize a relationship between the eigenvalues and generalized connectivity from a regular graph to a graph when $\kappa '<\delta$. Furthermore, we obtain a relationship between the eigenvalues and generalized connectivity of a graph when its girth is considered.

Theorem 1.5

Let l, k be integers such that $l\ge k \ge 2.$ Let G be a connected graph with $\mid V(G)\mid \ge k+l-1$, maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '$. If $\kappa '<\delta$ and

$$\begin{aligned} \lambda _{\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }(G) <\delta -1+\frac{2}{\delta +1}, \end{aligned}$$

then $\kappa _{l}(G)\ge k.$

For any integers $\delta$ and g with $\delta >0$ and $g\ge 3$, define $a=\lfloor \frac{g-1}{2}\rfloor ,$ and $n_{1}^{*}=n_{1}^{*}(\delta , g)$ as follows.

$$\begin{aligned} n_{1}^{*}=\left\{ \begin{array}{ll} 1+\delta +\sum _{i=2}^{a}(\delta -1)^{i},&{}\quad \text{ if } g=2a+1,\\ 2+2(\delta -1)^{a}+\sum _{i=2}^{a-1}(\delta -1)^{i},&{}\quad \text{ if } g=2a+2. \end{array} \right. \end{aligned}$$

(1.1)

Theorem 1.6

Let l, k be integers such that $l\ge k \ge 2.$ Let G be a connected graph with $\mid V(G)\mid \ge k+l-1$, maximum degree $\varDelta$, minimum degree $\delta (\ge 2)$, edge connectivity $\kappa '$ and girth $g=g(G)\ge 3$. If $\kappa '<\delta$ and

$$\begin{aligned} \lambda _{\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }(G) <\delta -\frac{\delta -1}{n_{1}^{*}}, \end{aligned}$$

where $n_{1}^{*}$ be defined as in (1.1), then $\kappa _{l}(G)\ge k$.

Many researchers studied a relationship between the eigenvalues and toughness of a regular graph. Alon [1] first obtained a lower bound of t(G).

Theorem 1.7

[1] For any connected d-regular graph G, $t(G)>\frac{1}{3}(\frac{d^{2}}{d\lambda +\lambda ^{2}}-1).$

Around the same time, Brouwer [4] improved the lower bound of t(G) and he [5] gave a conjecture about a lower bound of t(G).

Theorem 1.8

[4] For any connected d-regular graph G, $t(G)>\frac{d}{\lambda }-2.$

Conjecture 1.9

[5] For any connected d-regular graph G, $t(G)>\frac{d}{\lambda }-1.$

For the special case of toughness 1, Liu and Chen [17] improved the previous result of Brouwer.

Theorem 1.10

[17] For any connected d-regular graph G,  if

$$\begin{aligned} \lambda _{2}(G)< \left\{ \begin{array}{ll} d-1+\frac{3}{d+1}, &{}\quad \text{ if } d \text{ is } \text{ even },\\ d-1+\frac{2}{d+1}, &{}\quad \text{ if } d \text{ is } \text{ odd },\\ \end{array}\right. \end{aligned}$$

then $t(G)\ge 1.$

Recently, Cioabǎ and Wong [10] further improved the above result.

Theorem 1.11

[10] For any connected d-regular graph G,  if

$$\begin{aligned} \lambda _{2}(G)< \left\{ \begin{array}{ll} \frac{d-2+\sqrt{d^{2}+12}}{2}, &{}\quad \text{ if } d \text{ is } \text{ even },\\ \frac{d-2+\sqrt{d^{2}+8}}{2}, &{}\quad \text{ if } d \text{ is } \text{ odd },\\ \end{array}\right. \end{aligned}$$

then $t(G)\ge 1.$

Cioabǎ and Wong [10] also found the second largest eigenvalue condition for $t(G)=1$ in a bipartite connected d-regular graph.

Theorem 1.12

[10] For any bipartite connected d-regular graph G,  if

$$\begin{aligned} \lambda _{2}(G)< \left\{ \begin{array}{ll} \frac{d-2+\sqrt{d^{2}+12}}{2}, &{}\quad \text{ if } d \text{ is } \text{ even },\\ \frac{d-2+\sqrt{d^{2}+8}}{2}, &{}\quad \text{ if } d \text{ is } \text{ odd },\\ \end{array}\right. \end{aligned}$$

then $t(G)=1.$

Recently, Cioabǎ and Gu [8] further improved the above result.

Theorem 1.13

[8] For any bipartite connected d-regular graph G with $\kappa '<d$, if

$$\begin{aligned} \lambda _{\lceil \frac{d}{d-\kappa '}\rceil }(G)<d-\frac{d-1}{2d}, \end{aligned}$$

then $t(G)=1.$

In this paper, we generalize a relationship between the eigenvalues and toughness from a bipartite regular graph to a bipartite graph.

Theorem 1.14

Let G be a connected bipartite graph with maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '<\varDelta$. If

$$\begin{aligned} \lambda _{\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }(G)<\delta -\frac{\varDelta -1}{2\delta }, \end{aligned}$$

then $t(G)=1.$

Cioabǎ and Wong [10] also found the second largest eigenvalue condition for $t(G)\ge \tau ,$ where $\tau \le \kappa '/d$ is a positive number.

Theorem 1.15

[10] Let G be a connected d-regular graph with edge connectivity $\kappa '$ and $d\ge 3.$ Suppose that $\tau$ is a positive number such that $\tau \le \kappa '/d$. If

$$\begin{aligned} \lambda _{2}(G)<d-\tau d/(d+1), \end{aligned}$$

then $t(G)\ge \tau .$

Cioabǎ and Gu [8] found that the eigenvalue condition of Theorem 1.15 is not needed.

Theorem 1.16

[8] Let G be a connected d-regular graph with edge connectivity $\kappa '$. Then $t(G)\ge \frac{\kappa '}{d}.$

The following Theorem 1.17 presents a lower bound of toughness of a graph. This result generalizes Theorem 1.16 from a regular graph to a graph.

Theorem 1.17

Let G be a connected graph with edge connectivity $\kappa '$ and maximum degree $\varDelta$. Then $t(G)\ge \frac{\kappa '}{\varDelta }.$

In Fiedler’s fundamental work, he showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex connectivity and edge connectivity. Recently, there has been a lot of activity concerning the connections between the spectrum of a graph and the existence of spanning k-trees, generalized connectivity and toughness [2, 8,9,10,11, 13,14,15,16, 18, 19, 21]. In this paper, inspired by the idea of [8], we, respectively, study the relationship between the eigenvalues and the existence of spanning k-trees, generalized connectivity and toughness in a graph G with given some parameters.

In Sect. 2, some known lemmas and preliminary results are given. In Sect. 3, we mainly present a relationship between the eigenvalues and the existence of spanning k-trees in a graph G. In Sect. 4, we mainly consider a relationship between the eigenvalues and generalized connectivity in a graph G. And we also obtain a relationship between the eigenvalues and generalized connectivity of a graph with given girth. In Sect. 5, we mainly present a relationship between the eigenvalues and toughness in a bipartite graph G. And we obtain a lower bound of toughness in a graph G in terms of edge connectivity $\kappa '$ and maximum degree $\varDelta$.

2 Preliminaries

In this section, we shall give some former results which will be used in our arguments.

Lemma 2.1

[4] For a connected graph G with n vertices and m edges, then $\lambda _{1}(G)\ge \overline{d}$, where $\overline{d}=\frac{2m}{n}$ denotes average degree of G.

Lemma 2.2

[22] Let B and C be Hermitian matrices of order n, and let $1 \le i,j \le n$. Then

• (i) $\lambda _{i}(B)+\lambda _{j}(C) \le \lambda _{i+j-n}(B+C)$ if $i+j \ge n+1.$

• (ii) $\lambda _{i}(B)+\lambda _{j}(C) \ge \lambda _{i+j-1}(B+C)$ if $i+j \le n+1.$

Lemma 2.3

Let G be a graph on n vertices with minimum degree $\delta$. Then $q_{i}(G)\ge \delta +\lambda _{i}(G)$ for $1\le i\le n$.

Proof

By Lemma 2.2, we have $\lambda _{i}(A(G))+\lambda _{n}(D(G)) \le \lambda _{i}(Q(G))$ for $1\le i\le n.$ Thus, we have $q_{i}(G)\ge \delta +\lambda _{i}(G)$ for $1\le i\le n$. $\square$

Lemma 2.4

Let G be a graph on n vertices with maximum degree $\varDelta$. Then $\mu _{n+1-i}(G)\le \varDelta -\lambda _{i}(G)$ for $1\le i\le n.$

Proof

By Lemma 2.2, we have $\lambda _{i}(A(G))+\lambda _{n+1-i}(L(G)) \le \lambda _{1}(D(G))$ for $1\le i\le n.$ Thus, we have $\mu _{n+1-i}(G)\le \varDelta -\lambda _{i}(G)$ for $1\le i\le n$. $\square$

Let G be a graph with vertex set V(G). For two disjoint subsets X and Y of V(G), we use e(X, Y) to denote the number of edges in the set with one end in X and the other end in Y.

Lemma 2.5

[8] Let $S_{1},S_{2},\ldots ,S_{p}$ be disjoint subsets of V(G) with $e(S_{i}, S_{j} ) = 0$ for $i \ne j$. For $1 \le i \le p,$ let $G[S_{i}]$ denote the subgraph of G induced by $S_{i}$. Then

$$\begin{aligned} \lambda _{p}(G)\ge \lambda _{p}(G[\cup _{1\le i\le p} S_{i}])\ge \min _{1\le i\le p}\lambda _{1}(G[S_{i}]). \end{aligned}$$

Lemma 2.6

[12, 23] Let G be a connected graph and let $k\ge 2$. If for any $S \subseteq V(G)$, $c(G-S) \le (k-2)|S|+2,$ then G has a spanning k-tree.

Lemma 2.7

[20] Let G be a graph with minimum degree $\delta \ge 2$ and girth $g=g(G)\ge 3,$ and X be a vertex subset of G. Let $n_{1}^{*}=n_{1}^{*}(\delta , g)$ be defined as in (1.1). If $e(X, V(G)-X)<\delta ,$ then $|X|\ge n_{1}^{*}.$

3 Eigenvalues and Spanning k-Trees in a Graph

In this section, we prove Theorem 1.3, and give some corollaries that obviously follow from Theorem 1.3.

Proof of Theorem 1.3

We prove it by contradiction and assume that G does not have spanning k-trees for $k\ge 3.$ By Lemma 2.6, there exists a subset $X\subseteq V(G)$ such that

$$\begin{aligned} c(G-X)\ge (k-2)|X|+3. \end{aligned}$$

(3.1)

Let $x=|X|$, $t=c(G-X)$ and let $H_{1},H_{2},\ldots ,H_{t}$ be the components of $G-X.$ Let $n_{i}=|V (H_{i})|$ and let $s_{i}$ be the number of edges between $H_{i}$ and X for $1 \le i \le t.$ Then $s_{i}\ge \kappa '$ for $1 \le i \le t.$ As G is a connected graph with maximum degree $\varDelta$, we have $t\kappa '\le \sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x.$ By (3.1), $x \le (t-3)/(k-2).$ Thus, $t\kappa ' \le \varDelta (t-3)/(k-2)$, which implies that

$$\begin{aligned} t(\varDelta -(k-2)\kappa ')\ge 3\varDelta . \end{aligned}$$

(3.2)

Thus, $l=\varDelta -(k-2)\kappa '> 0,$ contrary to $l\le 0.$ This proves (i).

By (3.2), $t\ge \lceil \frac{3\varDelta }{l}\rceil .$ We claim that there are at least $\lceil \frac{3\varDelta }{l}\rceil$ indices i such that $s_{i}<\frac{\varDelta }{k-2}.$ Otherwise, there would be at most $\lceil \frac{3\varDelta }{l}\rceil -1$ indices i such that $s_{i}<\frac{\varDelta }{k-2}.$ In other words, there would be at least $t- \lceil \frac{3\varDelta }{l}\rceil +1$ indices i with $s_{i}\ge \frac{\varDelta }{k-2}.$ Thus,

$$\begin{aligned} \sum \limits ^{t}_{i=1}s_{i}&\ge \left( t-\left\lceil \frac{3\varDelta }{l}\right\rceil +1\right) \frac{\varDelta }{k-2}+\left( \left\lceil \frac{3\varDelta }{l}\right\rceil -1\right) \kappa '\\&=\frac{t\varDelta }{k-2}-\left( \left\lceil \frac{3\varDelta }{l}\right\rceil -1\right) \left( \frac{\varDelta }{k-2}-\kappa '\right) \\&>\frac{t\varDelta }{k-2}-\frac{3\varDelta }{l} \left( \frac{\varDelta }{k-2}-\kappa '\right) \\&=\frac{t\varDelta }{k-2}-\frac{3\varDelta }{k-2}\\&=\varDelta \frac{t-3}{k-2}\ge \varDelta x, \end{aligned}$$

contrary to $\sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x.$ This proves that there are at least $\lceil \frac{3\varDelta }{l}\rceil$ indices i such that $s_{i}<\frac{\varDelta }{(k-2)}.$

Without loss of generality, we may assume these indices are $1,2,\ldots ,\lceil \frac{3\varDelta }{l}\rceil .$ For $1 \le i \le \lceil \frac{3\varDelta }{l}\rceil ,$ we have $n_{i}\ge \delta +1.$ Otherwise, if $n_{i} \le \delta$ then $\delta n_{i}\le s_{i}+2|E(H_{i})|\le s_{i}+n_{i}(n_{i}-1)\le s_{i}+\delta (n_{i}-1),$ which implies $s_{i}\ge \delta .$ If $\varDelta \le (k-2)\delta$, we have $s_{i}\ge \delta \ge \frac{\varDelta }{(k-2)}$, contrary to $s_{i}<\frac{\varDelta }{(k-2)}.$

By Lemmas 2.1 and 2.5, we have

$$\begin{aligned} \lambda _{\lceil \frac{3\varDelta }{l}\rceil }(G)&\ge \min _{1\le i\le \lceil \frac{3d}{l}\rceil }\{\lambda _{1}(H_{i})\}\ge \min _{1\le i\le \lceil \frac{3d}{l}\rceil }\left\{ \frac{2m_{i}}{n_{i}}\right\} \\&\ge \min _{1\le i\le \lceil \frac{3d}{l}\rceil }\left\{ \frac{\delta n_{i}-s_{i}}{n_{i}}\right\} >\delta -\frac{\varDelta }{(k-2)(\delta +1)}, \end{aligned}$$

contrary to the assumption. This completes the proof. $\square$

Corollary 3.1

Let G be a connected graph with maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '$, and let $k \ge 3$. Let $l=\varDelta -(k-2)\kappa '$. Each of the following statements holds.

• (i) If $l \le 0,$ then G has a spanning k-tree.

• (ii) If $l > 0$, $\varDelta \le (k-2)\delta$ and $q_{\lceil \frac{3\varDelta }{l}\rceil }\le 2\delta -\frac{\varDelta }{(k-2)(\delta +1)},$ then G has a spanning k-tree.

Proof

It follows from Lemma 2.3 and Theorem 1.3.

Corollary 3.2

Let G be a connected graph on n vertices with maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '$, and let $k \ge 3$. Let $l=\varDelta -(k-2)\kappa '$. Each of the following statements holds.

• (i) If $l \le 0,$ then G has a spanning k-tree.

• (ii) If $l > 0$, $\varDelta \le (k-2)\delta$ and $\mu _{n+1-\lceil \frac{3\varDelta }{l}\rceil }\ge \varDelta - \delta +\frac{\varDelta }{(k-2)(\delta +1)},$ then G has a spanning k-tree.

Proof

It follows from Lemma 2.4 and Theorem 1.3. $\square$

4 Eigenvalues and Generalized Connectivity in a Graph

In this section, we mainly prove Theorems 1.5 and 1.6, and give some corollaries that obviously follow from Theorem 1.5.

Proof of Theorem 1.5

We prove it by contradiction and assume that $\kappa _{l}(G)<k$. By definition, there exists a subset $X \subset V(G)$ such that $|X|\le k-1$ and $c(G-X)\ge l$. Let $x=|X|$, $t=c(G-X)$ and let $H_{1},H_{2},\ldots ,H_{t}$ be the components of $G-X$. For $1 \le i \le t,$ let $n_{i}=|V (H_{i})|$ and let $s_{i}$ be the number of edges between $H_{i}$ and X. Then $s_{i}\ge \kappa '$ for $1 \le i \le t$. Since G is a graph, we have $\sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x.$

As $l\kappa '\le t\kappa ' \le \sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x \le \varDelta (k-1),$ we have $l\delta -l\kappa '\ge l\delta -\varDelta (k-1)$. Thus, we have $l\ge \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}$. We claim that there are at least $\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil$ indices i such that $s_{i}<\delta$. Otherwise, there would be at most $\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil -1$ indices i such that $s_{ i }<\delta .$ In other words, there would be at least $t-\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil +1$ indices i with $s_{i}\ge \delta .$ Thus,

$$\begin{aligned} \sum \limits ^{t}_{i=1}s_{i}&\ge \left( t-\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil +1\right) \delta +\left( \lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil -1\right) \kappa '\\&=t\delta -\left( \lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil -1\right) (\delta -\kappa ')\\&>t\delta -\left( \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\right) (\delta -\kappa ')\\&=t\delta -(l\delta -\varDelta (k-1))\\&=(t-l)\delta +(k-1)\varDelta \ge \varDelta x, \end{aligned}$$

contrary to $\sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x.$ Thus, there are at least $\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil$ indices i such that $s_{i}<\delta .$

Without loss of generality, we may assume these indices are $1,2,\ldots ,\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil .$ For $1\le i \le \lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil ,$ we have $n_{i}\ge \delta +1.$ Otherwise, if $n_{i} \le \delta ,$ then $\delta n_{i}\le s_{i}+2|E(H_{i})|\le s_{i}+n_{i}(n_{i}-1)\le s_{i}+\delta (n_{i}-1),$ which implies $s_{i}\ge \delta ,$ contrary to $s_{i}<\delta .$

By Lemmas 2.1 and 2.5, we have

$$\begin{aligned} \lambda _{\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }(G)&\ge \min _{1\le i\le \lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }\{\lambda _{1}(H_{i})\}\\&\ge \min _{1\le i\le \lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }\left\{ \frac{2m_{i}}{n_{i}}\right\} \\&\ge \min _{1\le i\le \lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }\left\{ \frac{\delta n_{i}-s_{i}}{n_{i}}\right\} \\&\ge \delta -\frac{\delta -1}{\delta +1}\\&=\delta -1+\frac{2}{\delta +1}, \end{aligned}$$

contrary to the assumption. This completes the proof.

Proof of Theorem 1.6

Using the similar method as that used in Theorem 1.5, it follows by Lemma 2.7.

Corollary 4.1

Let l, k be integers such that $l\ge k \ge 2.$ Let G be a connected graph with $\mid V(G)\mid \ge k+l-1$, maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '$. If $\kappa '<\delta$ and

$$\begin{aligned} q_{\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }(G) <2\delta -1+\frac{2}{\delta +1}, \end{aligned}$$

then $\kappa _{l}(G)\ge k.$

Proof

It follows from Lemma 2.3 and Theorem 1.5.

Corollary 4.2

Let l, k be integers such that $l\ge k \ge 2.$ Let G be a connected graph with $n=\mid V(G)\mid \ge k+l-1$, maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '$. If $\kappa '<\delta$ and

$$\begin{aligned} \mu _{n+1-\lceil \frac{l\delta -\varDelta (k-1)}{\delta -\kappa '}\rceil }(G)> \varDelta -\delta +1-\frac{2}{\delta +1}, \end{aligned}$$

then $\kappa _{l}(G)\ge k.$

Proof

It follows from Lemma 2.4 and Theorem 1.5. $\square$

Similarly, we also can obtain a relationship between the (signless) Laplacian eigenvalues and generalized connectivity of a graph with given girth. Here we omit them.

5 Eigenvalues and Toughness in a Bipartite Graph

In this section, we mainly study a relationship between the toughness of a bipartite graph G and its eigenvalues. We mainly prove Theorems 1.14 and 1.17, and give some corollaries that obviously follow from Theorem 1.14.

Theorem 5.1

Let G be a bipartite graph. Then $t(G)\le 1.$

Proof

Let X, Y be the set of vertices of two parts of a bipartite graph G. Without loss of generality, assume $|X|\le |Y|.$ It is easy to see $|Y|=c(G-X).$ By definition of toughness, we have $t(G)\le 1.$

Proof of Theorem 1.14

We prove it by contradiction and assume that $t(G)\ne 1$. By Theorem 5.1, we have $t(G)<1.$ By definition, there exists a subset $X \subset V(G)$ such that $|X|/c(G-X)<1$. Let $x=|X|$, $t=c(G-X)$ and let $H_{1},H_{2},\ldots ,H_{t}$ be the components of $G-X$. For $1 \le i \le t,$ let $n_{i}=|V (H_{i})|$ and let $s_{i}$ be the number of edges between $H_{i}$ and X. Then $x<t$ and $s_{i}\ge \kappa '$ for $1 \le i \le t$. Since G is a graph, we have $\sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x.$

As $t\kappa ' \le \sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x \le \varDelta (t-1),$ we have $t(\varDelta -\kappa ')\ge \varDelta$, i.e., $t\ge \frac{\varDelta }{\varDelta -\kappa '}$. We claim that there are at least $\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil$ indices i such that $s_{i}< \varDelta .$ Otherwise, there would be at most $\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil -1$ indices i such that $s_{i}< \varDelta .$ In other words, there would be at least $t-\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil +1$ indices i such that $s_{i}\ge \varDelta .$ Thus, we have

$$\begin{aligned} \sum \limits ^{t}_{i=1}s_{i}&\ge \left( t-\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil +1\right) \varDelta +\left( \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil -1\right) \kappa '\\&=t\varDelta -\left( \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil -1\right) (\varDelta -\kappa ')\\&>t\varDelta -\frac{\varDelta }{\varDelta -\kappa '}(\varDelta -\kappa ')\\&=\varDelta (t-1), \end{aligned}$$

contrary to $\sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta (t-1).$ Thus, there are at least $\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil$ indices i such that $s_{i}<\varDelta .$

Since each $H_{i}$ is a bipartite graph, we have $m_{i}\le \frac{n_{i}^{2}}{4}$ for $1 \le i \le t.$ By $2m_{i}\ge \delta n_{i}-s_{i}$, we have $\frac{n_{i}^{2}}{2} \ge 2m_{i}\ge \delta n_{i}-s_{i} \ge \delta n_{i}-\varDelta +1$ for $1 \le i \le \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil$. Hence $n_{i}^{2}-2\delta n_{i}+2\varDelta -2\ge 0.$ It is easy to see $n_{i}\ge 2\delta$ for $1 \le i \le \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil$. Thus, by Lemma 2.5, we have

$$\begin{aligned} \lambda _{\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }&\ge \min _{1\le i \le \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }\{\lambda _{1}(H_{i})\}\ge \min _{1\le i \le \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }\left\{ \frac{2|E(H_{i})|}{n_{i}}\right\} \\&\ge \min _{1\le i \le \lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }\left\{ \frac{\delta n_{i}-s_{i}}{n_{i}}\right\} \ge \delta -\frac{\varDelta -1}{2\delta }, \end{aligned}$$

contrary to the assumption. This completes the proof.

Corollary 5.2

Let G be a connected bipartite graph with maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '<\varDelta$. If

$$\begin{aligned} q_{\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }(G)<2\delta -\frac{\varDelta -1}{2\delta }, \end{aligned}$$

then $t(G)=1.$

Proof

It follows from Lemma 2.3 and Theorem 1.14. $\square$

Corollary 5.3

Let G be a connected bipartite graph on n vertices with maximum degree $\varDelta$, minimum degree $\delta$ and edge connectivity $\kappa '<\varDelta$. If

$$\begin{aligned} \mu _{n+1-\lceil \frac{\varDelta }{\varDelta -\kappa '}\rceil }(G)> \varDelta -\delta +\frac{\varDelta -1}{2\delta }, \end{aligned}$$

then $t(G)=1.$

Proof

It follows from Lemma 2.4 and Theorem 1.14. $\square$

In the following, we give the proof of Theorem 1.17.

Proof of Theorem 1.17

Suppose that X is a vertex cut of G. Let $x=|X|$, $t=c(G-S)$ and let $H_{1},H_{2},\ldots ,H_{t}$ be the components of $G-X.$ Let $n_{i}=|V (H_{i})|$ and let $s_{i}$ be the number of edges between $H_{i}$ and X for $1 \le i \le t.$ Then $s_{i}\ge \kappa '$ for $1 \le i \le t.$ As G is a connected graph with maximum degree $\varDelta$, $\sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x.$ Thus, $t\kappa '\le \sum \nolimits ^{t}_{i=1}s_{i}\le \varDelta x,$ which implies that $\frac{x}{t}\ge \frac{\kappa '}{\varDelta }.$ Hence, $t(G) \ge \frac{\kappa '}{\varDelta }$. $\square$