Amitava Bhattacharya, Shmuel Friedland, Uri N. Peled
In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi- Hoffman conjecture for general graphs, and prove the conjecture in some special cases.
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On non-bipartite graphs with integral signless Laplacian eigenvalues at most 6
该研究完成了符号拉普拉斯特征值为不超过6的整数的连通非二部图的完整分类工作，属于组合数学领域的图论研究。
Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first eigenvectors of $G$. In this paper we give some numerical and structural properties about the first eigenvectors of $G$ which contains an odd-bipartite branch, and investigate how the least eigenvalue of $G$ changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue att…
In this paper we study the local law for eigenvalues of large random regular bipartite graphs with degree growing arbitrarily fast. We prove that the empirical spectral distribution of the adjacency matrix converges to a scaled down copy of the Marchenko - Pastur distribution on intervals of short length.
Let $G$ be a connected non-bipartite graph on $n$ vertices with domination number $γ\le \frac{n+1}{3}$. We investigate the least eigenvalue of the signless Laplacian of $G$, and present a lower bound for such eigenvalue in terms of the domination number $γ$.
In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of signless Laplacian of a graph in terms of the number of pendent vertices.
Eigenvalue Interlacing of Bipartite Graphs and Construction of Expander Code using Vertex-split of a Bipartite Graph
该研究证明了二分图邻接矩阵与二分商矩阵的特征值交错关系及特征值界，提出二分图顶点分裂法构造渐近良好Expander码，证明了相关连通性与扩张性结论。
It is proved that the median eigenvalues of every connected bipartite graph $G$ of maximum degree at most three belong to the interval $[-1,1]$ with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$. Moreover, if $G$ is not isomorphic to the Heawood graph, then a positive fraction of its median eigenvalues lie in the interval $[-1,1]$. This surprising result has been motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry.
The smallest normalized signless $\infty$-Laplacian eigenvalue for non-bipartite connected graphs
本文研究了非二部连通图的最小规范化无符号∞-拉普拉斯特征值，证明其等于加权无符号关联矩阵广义逆的最小∞-范数的倒数，并举例说明。
For a graph $G$ of order $n$ and with eigenvalues $λ_1\geqslant\cdots\geqslantλ_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|λ_{\lfloor(n+1)/2\rfloor}|, |λ_{\lceil(n+1)/2\rceil}|\right\}.$ We show that for every connected bipartite graph $G$ with maximum degree $Δ\geqslant3$, $R(G)\leqslant\sqrt{Δ-2}$ unless $G$ is the the incidence graph of a projective plane of order $Δ-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index.
In 1974 Cvetković and Simić showed which graphs $G$ are the bipartite complements of line graphs. In 2002 Borovićanin showed which line graphs $L(H)$ have third largest eigenvalue $λ_3\leq0$. Our first observation is that two of the graphs Borovićanin found are the complements of two of the graphs found by Cvetković and Simić. Using the Courant-Weyl inequalities we show why this is and reprove the result of Borovićanin, highlighting some features of the graphs found by both.
Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. The adjacency matrix of $Γ=(G, σ)$ is a square matrix $A(Γ)=A(G, σ)=\left(a_{i j}^σ\right)$, where $a_{i j}^σ=σ\left(v_{i} v_{j}\right) a_{i j}$. In this paper, we determine the eigenvalues of the signed complete bipartite graphs. Let $(K_{p, q},σ)$, $p\leq q$, be a signed complete bipartite graph with bipartition $(U_p, V_q)$, where $U_p=\{u_1,u_2,\ldots,u_p\}$ and $V_q=\{v_1,v_2,\ldots,v_q\}$. Let $(K_{p, q},σ)[U_r\cup V_s]$, $r\leq p$ and $s\leq q $, be an induced signed subgraph on minimum vertices $r+s$, …
In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $θ_{\min}\leq -αk$, where $0<α<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $θ_{\min}$ compared with $k$. In particular, we will show that if $θ_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $θ_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.
A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Non-bipartite distance-regular graphs with a small smallest eigenvalue, Electronic J. Combin. 26(2) (2019), P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.
A recent result of one of the authors says that every connected subcubic bipartite graph that is not isomorphic to the Heawood graph has at least one, and in fact a positive proportion of its eigenvalues in the interval [-1,1]. We construct an infinite family of connected cubic bipartite graphs which have no eigenvalues in the open interval (-1,1), thus showing that the interval [-1,1] cannot be replaced by any smaller symmetric subinterval even when allowing any finite number of exceptions. Similar examples with vertices of larger degrees are considered and it is also shown that their eigenva…
We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only $C_6$. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.
Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues
本文研究图的最小不同特征值数q(G)，证明补图边数不超过⌊n/2⌋-1时q(G)=2，并针对二部补图情形验证了q(G)=2的猜想，同时刻画了边数为n-2时q(G)>2的图。
We study the spectrum of the normalized Laplace operator of a connected graph $Γ$. As is well known, the smallest nontrivial eigenvalue measures how difficult it is to decompose $Γ$ into two large pieces, whereas the largest eigenvalue controls how close $Γ$ is to being bipartite. The smallest eigenvalue can be controlled by the Cheeger constant, and we establish a dual construction that controls the largest eigenvalue. Moreover, we find that the neighborhood graphs $Γ[l]$ of order $l\geq2$ encode important spectral information about $Γ$ itself which we systematically explore. In particular, t…
A pseudo $(v,\, k,\, \la)$-design is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}=\{B_1,...,B_{v-1}\}$ is a collection of $k$-subsets (blocks) of $X$ such that each two distinct $B_i, B_j$ intersect in $\la$ elements; and $0\le\la <k \le v-1$. We use the notion of pseudo designs to characterize graphs of order $n$ whose (adjacency) spectrum contains a zero and $\pmθ$ with multiplicity $(n-3)/2$ where $0<θ\le\sqrt{2}$. Meanwhile, partial results confirming a conjecture of O. Marrero on characterization of pseudo $(v,\, k,\, \la)$-designs are obtained.
Extremal eigenvalues of outerplanar graphs
本文研究外平面图的极值特征值，通过刻画极大二部外平面图的结构，完全确定了阶数至少55的二部外平面图的最大谱半径和一般外平面图的最小最小特征值。
We analyse the eigenvalues of Erdös--Rényi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=Ω(\sqrt{n_{1}p}\log^{3}(n_{1})),$ $n_{2}p=Ω(\sqrt{n_{2}p}\log^{3}(n_{2})),$ and let $G\sim G(n_{1},n_{2},p)$. We show that with probability tending to $1$ as $n_{1}$ tends to infinity: $$μ_{2} (A(G))\leq 2[1+o(1)](\sqrt{n_{1}p}+\sqrt{n_{2}p}+\sqrt{(n_{1}+n_{2})p}).$$
Let $X=(V, E)$ be a finite regular graph and $H_t(u, v), \, u, v \in V$, the heat kernel on $X$. We prove that, if the graph $X$ is bipartite and has four distinct Laplacian eigenvalues, the ratio $H_t(u, v)/H_t(u, u), \, u, v \in V,$ is monotonically non-decreasing as a function of $t$. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.
In this paper, we consider the Hessian matrices $H_Γ$ of the complete and complete bipartite graphs, and the special value of $\tilde H_Γ$ at $x_{i}=1$ for all $x_{i}$. We compute the eigenvalues of $\tilde H_Γ$. We show that one of them is positive and that the others are negative. In other words, the metric with respect to the symmetric matrix $\tilde H_Γ$ is Lorentzian. Hence those Hessian $\det (H_Γ)$ are not identically zero. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the graphic matroids of the complete and complete bipartit…
Global eigenvalue fluctuations of random biregular bipartite graphs
本文研究随机双正则二部图的全局特征值涨落，通过泊松逼近圈数与循环非回溯游走数得到全变差距离界，并证明度比趋于无穷时的半圆律，进而将结果推广到随机正则超图的邻接矩阵。
Let $G$ be a graph of order $n$ and $μ$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $μ$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $μ$, and the vertex subset $X=V(G-H)$ is called a star set for $μ$ in $G$. The study of star complements and star sets provides a strong link between graph structure and linear algebra. In this paper, we study the regular graphs with $K_{t,s}\ (s\geq t\geq 2)$ as a star complement for an eigenvalue $μ$, especially, characterize the case of $t=3$ completely, obtain some properties when $t=s$, a…
Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes
本文将二部图基于奇环的刻画推广到可去定向单纯复形，刻画其下对偶图的环条件，并揭示奇环数量与最大 Hodge Laplacian 特征值及高维二部划分的关系。
The main eigenvalues of a graph $G$ are those eigenvalues of the $(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct product $G\times K_2$. The main eigenspace of $\mathbf A$ is generated by the principal main eigenvectors and is the same as the image of the walk matrix. A hierarchy of properties of pairs of graphs is established in view of their CDC's, walk matrices, main eigenvalues, eigenvectors and eigenspaces. We determine by algorithm that there are 32 pairs of non-isomorphic graphs on a…
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
For a graph $G = (V, E)$, the $γ$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $γ$-sets of $G$, and two $γ$-sets are adjacent if they differ by a single vertex and the two different vertices are adjacent in $G$. An open question in $γ$-graphs is whether every bipartite graph is the $γ$-graph of some bipartite graph. We answer this question in the negative by demonstrating that $K_{2, 3}$ is not the $γ$-graph of any bipartite graph.
In this paper, we study the non-bipartite distance-regular graphs with valency k and having a smallest eigenvalue at most -k/2.
Let $G$ be a simple graph or hypergraph, and let $A(G),L(G),Q(G)$ be the adjacency, Laplacian and signless Laplacian tensors of $G$ respectively. The largest $H$-eigenvalues (resp., the spectral radii) of $L(G),Q(G)$ are denoted respectively by $λ_{\max}^L(G), λ_{\max}^Q(G)$ (resp., $ρ^L(G), ρ^Q(G)$). For a connected non-bipartite simple graph $G$, $λ_{\max}^L(G)=ρ^L(G) < ρ^Q(G)$. But this does not hold for non-odd-bipartite hypergraphs. We will investigate this problem by considering a class of generalized power hypergraphs $G^{k,\frac{k}{2}}$, which are constructed from simple connected grap…
We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of `irregular Ramanujan' graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c,d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by sq…
We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We construct some families of bipartite signed graphs with only two distinct eigenvalues. This leads to constructing infinite families of regular bipartite Ramanujan graphs.
Constructing cospectral graphs by unfolding non-bipartite graphs
本文将二部图的展开操作推广到非二部图，利用自反二部图、半自反二部图和多部图的展开构造邻接矩阵同谱但不同构的图。
This paper deals with the problem of finding totally antimagic total labelings of complete bipartite graphs. We prove that complete bipartite graphs are totally antimagic total graphs. We also show that the join of complete bipartite graphs with one vertex is a totally antimagic total graph
The existence of biregular spanning subgraphs in bipartite graphs via spectral radius
本文利用谱半径证明二部图中存在双正则二部生成子图，并进一步研究度数受限生成树与边不相交生成树的存在条件。
The graph packing problem is a well-known area in graph theory. We consider a bipartite version and give almost tight conditions on the packability of two bipartite sequences.
In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph $G$ and the same invariant in the complement $G^c$ of $G$ is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph $G=G[X,Y]$ with bipartition ($X,Y$), its bipartite complementary graph $G^{bc}$ is a bipartite graph with $V(G^{bc})…
We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.
Countable homogeneous ordered bipartite graphs
本文对所有可数齐次有序二部图进行了完整分类。
We show that the number of labelled $P_7$-free bipartite graphs with $n$ vertices grows as $n^{Θ(n)}$. This resolves an open problem posed by Allen [P. Allen, Forbidden induced bipartite graphs. J. Graph Theory 60 (2009), no. 3, 219--241.], and completes the description of speeds of monogenic classes of bipartite graphs. Our solution is based on a new decomposition scheme of bipartite graphs, which is of independent interest.
The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while holding the other two fixed. Here we focus in {\em bipartite Moore graphs\/}, that is, bipartite graphs attaining the optimum order, fixed either the degree/diameter or degree/girth. The fact that there are very few bipartite Moore graphs suggests the relaxation of some of the constraints implied by the bipartite Moore bound. First we deal with {\em local bipartit…
The minimum size of 2-connected chordal bipartite graphs
本文研究2-连通弦二分图的边数下界，确定了任意给定阶数下此类图的最小尺寸。
The Full P-vertex Problem and Perfect Matchings for Bipartite Graphs
研究二分图中性质(P)与完美匹配的关系，证明性质(P)等价于完美匹配存在的条件，并应用于多类二分图族。
The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.
In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2Δ(G)$, where $Δ(G)$ is the largest degree of $G$, if and only if $G$ is regular. Thus the largest Laplacian H-eigenvalue of $G$, reaches the same upper bound, if and only if $G$ is regular and odd-bipartite. We show that an $s$-cycle $G$, as a $k$-uniform hypergraph, where $1 \le s \le k-1$, is regular if and only if there is a positive integer $q$ such that $k=q(k-s)$. We show that an even-uniform $s$-path and an even-uniform non-regular…
The Estrada index of a graph $G$ is defined as $EE(G)=\sum_{i=1}^ne^{λ_i}$, where $λ_1,$ $ λ_2,\ldots, λ_n$ are the eigenvalues of the adjacency matrix of $G$. In this paper, we characterize the unique bipartite graph with maximum Estrada index among bipartite graphs with given matching number and given vertex-connectivity, edge-connectivity, respectively.
Outerplane bipartite graphs with isomorphic resonance graphs
本文研究了2-连通外平面二部图的同构共振图，给出了其结构刻画，并分别用共振有向图和内部对偶的局部结构提供了两种等价表征。
We give a simple proof that every $n$-vertex graph $d$-regular graph that does not contain a fixed bipartite graph as a subgraph has an induced matching of size $Ω((n/d)(\log d))$.
A graph $X$ is said to be chordal bipartite if it is bipartite and contains no induced cycle of length at least $6$. It is proved that if $X$ does not contain bipartite claw as an induced subgraph, then the Weisfeiler-Leman dimension of $X$ is at most $3$. The proof is based on the theory of coherent configurations.
In this paper, we investigate odd graceful graph, odd strongly harmonious graph, bipartite graph and their relationship. We proved following results: (1) if G is odd strongly harmonious graph, then G is odd graceful graph ;(2) if G is bipartite odd graceful graph, then G is odd strongly harmonious graph.
In this paper, we completely characterize the niche graphs of bipartite tournaments and find their interesting properties.
A graph is {\em near-bipartite} if its vertex set can be partitioned into an independent set and a set that induces a forest. It is clear that near-bipartite graphs are $3$-colorable. In this note, we show that planar graphs without cycles of lengths in $\{4, 6, 8\}$ are near-bipartite.
We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain $K_{3,3}$ as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite $(2,2)$-Laman graphs --- a certain family of graphs that contains all maximal bipartite planar graphs.
Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance in graphs) to find a formula for the number of spanning trees in the nearly complete bipartite graph $G(m,n,p)=K_{m,n}-pK_2$ $(p\leq \min\{m,n\})$, which extends a recent result by Ye and Yan who obtained the effective resistances and the number of spanning trees in $G(n,n,p)$. As a corollary, we obtain the Kirchhoff index of $G(m,n,p)$ which exte…
It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.
In this paper, we completely characterize the $m$-step competition graph of a bipartite tournament for any integer $m \ge 2$. In addition, we compute the competition index and the competition period of a bipartite tournament.
We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this context, the bipartite Moore bound $\M^b(d,D)$ represents a general upper bound for $\N^b(d,D)$. Bipartite graphs of order $\M^b(d,D)$ are very rare, and determining $\N^b(d,D)$ still remains an open problem for most $(d,D)$ pairs. This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on bipartite $(d,D,-4)$-graphs (that is, bipartite gr…
We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Δ \geq 2 and D \geq 2, find the maximum number Nb(Δ,D) of vertices in a bipartite graph of maximum degree Δ and diameter D. In this context, the Moore bipartite bound Mb(Δ,D) represents an upper bound for Nb(Δ,D). Bipartite graphs of maximum degree Δ, diameter D and order Mb(Δ,D), called Moore bipartite graphs, have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Δ \geq 2, diameter D \geq 2 and order Mb(Δ,D) - εwith small ε> 0, that is,…
In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is 2-extendable if and only if it is not a cycle. It is known that a non-bipartite Cayley (vertex-transitive) graph is 2-extendable when it is of minimum degree at least five \cite{sun}. We next classify all 2-extendable cubic non-bipartite Cayley graphs and obtain that: a cubic non-bipartite Cayley graph with girth $g$ is 2-extendable if and only if $g\geq 4$ and it does…
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.
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