Let $G$ be a simple graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. Let $ϕ(G; λ)=\det(λI-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a}_i(G)λ^{n-i}$ be the characteristic polynomial of $G$, where $\mathbf{a}_i(G)$ is called the $i$-th adjacency coefficient of $G$. Denote by $\mathfrak{B}_{n,m}$ the set of all connected graphs having $n$ vertices and $m$ edges. A bipartite graph $G$ is referred as bipartite optimal if $$\mathbf{a}_4(G)=min\{\mathbf{a}_4(H)|H\in \mathfrak{B}_{n,m}\}.$$ The value $min\{\mathbf{a}_4(H)|H\in \mathfrak{B}_{n,m}\}$ is called the minimal $4$-Sachs number in $\mathfrak{B}_{n,m}$, denoted by $\bar{\mathbf{a}}_4(\mathfrak{B}_{n,m})$. \vspace{2mm} For any given integer pair $(n,m)$, we in this paper investigate the bipartite optimal graphs. Firstly, we show that each bipartite optimal graph is a difference graph (see Theorem 10). Then we deduce some structural properties on bipartite optimal graphs. As applications of those properties, we determine all bipartite optimal $(n,m)$-graphs together with the corresponding minimal $4$-Sachs number for $n\ge 5$ and $n-1\le m\le 3(n-3)$. Finally, we express the problem of computing the minimal $4$-Sachs number as a class of combinatorial optimization problem, which relates to the partitions of positive integers.
We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $θ$-Hermitian adjacency matrices defined by an angle $θ\in (0, π]$. We show that this equivalence holds when, for example, an angle $θ$ is an algebraic number, while it breaks down for any angle $θ\in \mathbb{Q}π$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $θ\in \mathbb{Q}π$.
In [Steve Butler. A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390, 2010.], Butler constructed a family of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. In this article, we extend this construction for generating larger classes of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. Also, we provide a couple of constructions of non-bipartite graphs, which are cospectral for the adjacency matrices but not necessaril…
Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in …
Let $X$ be bipartite mixed graph and for a unit complex number $α$, $H_α$ be its $α$-hermitian adjacency matrix. If $X$ has a unique perfect matching, then $H_α$ has a hermitian inverse $H_α^{-1}$. In this paper we give a full description of the entries of $H_α^{-1}$ in terms of the paths between the vertices. Furthermore, for $α$ equals the primitive third root of unity $γ$ and for a unicyclic bipartite graph $X$ with unique perfect matching, we characterize when $H_γ^{-1}$ is $\pm 1$ diagonally similar to $γ$-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a n…
Let $G$ be a graph with order $n$ and adjacency matrix $\mathbf{A}(G)$. The adjacency polynomial of $G$ is defined as $ϕ(G;λ) =det(λ\mathbf{I}-\mathbf{A}(G))=\sum_{i=0}^n\mathbf{a_i}(G)λ^{n-i}$. Hereafter, $\mathbf{a}_i(G)$ is called the $i$-th adjacency coefficient of $G$. Denote by $\mathfrak{G}_{n,m}$ the set of all connected graphs having $n$ vertices and $m$ edges. A graph $G$ is said $4$-Sachs minimal if $$\mathbf{a}_4(G)=min\{\mathbf{a}_4(H)|H\in \mathfrak{G}_{n,m}\}.$$ The value $min\{\mathbf{a}_4(H)|H\in \mathfrak{G}_{n,m}\}$ is called the minimal $4$-Sachs number in $\mathfrak{G}_{n,…
In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph $G$, we construct a generalized power of $G$, denoted by $G^{k,s}$, which is obtained from $G$ by blowing up each vertex into a $k$-set and each edge into a $(k-2s)$-set, where $s \le k/2$. When $s < k/2$, $G^{k,s}$ is always odd-bipartite. We show that $G^{k,{k \over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and find that $G^{k,{k \over 2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as $G$. So the results involving the adjacency or si…
The generalized power of a simple graph $G$, denoted by $G^{k,s}$, is obtained from $G$ by blowing up each vertex into an $s$-set and each edge into a $k$-set, where $1 \le s \le \frac{k}{2}$. When $s < \frac{k}{2}$, $G^{k,s}$ is always odd-bipartite. It is known that $G^{k,{k \over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and $G^{k,{k \over 2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as $G$. In this paper, we prove that, regardless of multiplicities, the $H$-spectrum of $\A(G^{k,\frac{k}{2}})$ (respectively, $\Q(G^{k,\frac{k}{2}})$) con…
The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ via studying spectral properties of their adjacency matrices. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E$ the adjacency matrix of a union of vertex-disjoin…
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.
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.
A square (0,1)-matrix X of order n > 0 is called fully indecomposable if there exists no integer k with 0 < k < n, such that X has a k by n-k zero submatrix. A stable set of a graph G is a subset of pairwise nonadjacent vertices. The stability number of G, denoted by $α(G)$, is the cardinality of a maximum stable set in G. A graph is called $α$-stable if its stability number remains the same upon both the deletion and the addition of any edge. We show that a connected bipartite graph has exactly two maximum stable sets that partition its vertex set if and only if its reduced adjacency matrix i…
It can be shown that any symmetric $(0,1)$-matrix $A$ with $\tr A = 0$ can be interpreted as the adjacency matrix of a simple, finite graph. The square of an adjacency matrix $A^2=(s_{ij})$ has the property that $s_{ij}$ represents the number of walks of length two from vertex $i$ to vertex $j$. With this information, the motivating question behind this paper was to determine what conditions on a matrix $S$ are needed to have $S=A(G)^2$ for some graph $G$. Structural results imposed by the matrix $S$ include detecting bipartiteness or connectedness and counting four cycles. Row and column sums…
To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A^{-1} have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree.
Natural join of $di-bigraphs$ that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their $Hasse$ digraphs called $KoDAGs$. $KoDAGs$ are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. $KoDAGs$ appear to be distinguished family of $Ferrers$ digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called $di-bicliques$. These digraphs serve to represent faithfully correspon…
It is well known that spectral Turán type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Turán type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Denote by $λ_1$ and $λ_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. …
Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers $n$, the $n$-vertex graph $G$ that maximizes spread is the join of a clique and an independent set, with $\lfloor 2n/3 \rfloor$ and $\lceil n/3 \rceil$ vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all…
A \emph{directional labeling} of an edge $\emph{uv}$ in a graph $G=(V,E)$ by an ordered pair $ab$ is a labeling of the edge $uv$ such that the label on $uv$ in the direction from $u$ to $v$ is $\ell(uv)=ab$, and $\ell(vu)=ba$. New characterizations of 2-colorable (bipartite) and 3-colorable graphs are obtained in terms of directional labeling of edges of a graph by ordered pairs $ab$ and $ba$. In addition we obtain characterizations of 2-colorable and 3-colorable graphs in terms of matrices called directional adjacency matrices.
A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron-Frobenius theory, we establish a …
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.
Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are more close to bipartite simple graphs from the viewpoint of both structure and spectrum. A hypergraph is called minimal non-odd-transversal if it is non-odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and …
For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $λ_{1}\geqλ_{2}\geq\cdots\geqλ_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|λ_{j}|$. Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of this concept in terms of the contributions of even and odd walks, and consequently on the contribution of subgraphs, is not yet known. Here, we find such interpretation and prove that th…
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.
The permanental polynomial of a graph $G$ is $π(G,x)\triangleq\mathrm{per}(xI-A(G))$. From the result that a bipartite graph $G$ admits an orientation $G^e$ such that every cycle is oddly oriented if and only if it contains no even subdivision of $K_{2,3}$, Yan and Zhang showed that the permanental polynomial of such a bipartite graph $G$ can be expressed as the characteristic polynomial of the skew adjacency matrix $A(G^e)$. In this paper we first prove that this equality holds only if the bipartite graph $G$ contains no even subdivision of $K_{2,3}$. Then we prove that such bipartite graphs …
We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem we consider a balanced biclique with the same number of vertices in both parts and arbitrary edge weights. In the other two problems we are dealing with unbalanced subgraphs of maximum and minimum weight with nonnegative edges. All three problems are established to be NP-hard. We study the polytopes and the cone decompositions of these problems and their 1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope …
Let $Γ$ denote a finite, undirected, connected graph, with vertex set $X$. Fix a vertex $x \in X$. Associated with $x$ is a certain subalgebra $T=T(x)$ of ${\rm Mat}_X(\mathbb C)$, called the subconstituent algebra. The algebra $T$ is semisimple. Hora and Obata introduced a certain subalgebra $Q \subseteq T$, called the quantum adjacency algebra. The algebra $Q$ is semisimple. In this paper we investigate how $Q$ and $T$ are related. In many cases $Q=T$, but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible $T$-modules. We show that the…
For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its isoperimetric constant.
This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction recovers the classical zig-zag product of graphs when the permutation is an involution, and it is in fact more general. We start by discussing necessary and sufficient conditions on the permutation and on the adjacency matrix of a graph to guarantee their composition to represent an adjacency matrix of a graph, then we focus our attention on the cases in which …
This paper is the second chapter of three of the author's undergraduate thesis. In this paper, we consider the random matrix ensemble given by $(d_b, d_w)$-regular graphs on $M$ black vertices and $N$ white vertices, where $d_b \in [N^γ, N^{2/3 - γ}]$ for any $γ> 0$. We simultaneously prove that the bulk eigenvalue correlation statistics for both normalized adjacency matrices and their corresponding covariance matrices are stable for short times. Combined with an ergodicity analysis of the Dyson Brownian motion in another paper, this proves universality of bulk eigenvalue correlation statistic…
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…
Given a graph $G$, let $G^σ$ be an oriented graph of $G$ with the orientation $σ$ and skew-adjacency matrix $S(G^σ)$. Then the spectrum of $S(G^σ)$ is called the skew-spectrum of $G^σ$, denoted by $Sp_S(G^σ)$. It is known that a graph $G$ is bipartite if and only if there is an orientation $σ$ of $G$ such that $Sp_S(G^σ)=iSp(G)$. In [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electron. J. Combin. 20(2013), #P19], Cui and Hou conjectured that such orientation of a bipartite graph is unique under switching-equivalence. In this paper, we prove that the conjecture is true…
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…
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_Σ(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}η, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_Σ(G,S)$ and $η= 2…
Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted $\nabla^{\mathrm{PQ}}_G$ and which is the focus of this work, encodes rich combinatorial information about power-flow solutions in sparse power networks that are studied in electric engineering. Of particular importance is the normalized volume of such an adjacency polytope, which provides an upper bound on the number of distinct power-flow solutions. In this article we show that the problem of computing normalized volumes for $\nabla^{\mathrm{PQ}}_G$…
The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in [European J. Combin. 95 (2021) 103328], showed that the minimum algebraic connectivity of cubic connected graphs on $2n$ vertices is $(1+o(1))\frac{π^{2}}{2n^{2}}$, which is attained on non-bipartite graphs. Motivated by the above result, we in this paper investigate the algebraic connectivity of cubic bipartite graphs. We prove that the minimum algebraic co…
In this work we study the null space of bipartite graphs without cycles of length multiple of $4$, and its relation to structural properties. We decompose them into two subgraphs: $C_N(G)$ and $C_S(G)$. $C_N(G)$ has perfect matching and its adjacency matrix is nonsingular. $C_S(G)$ has a unique maximum independent set and the dimension of its null space equals the dimension of the null space of $G$. Even more, we show that the fundamental spaces of $G$ are the direct sum of the fundamental spaces of $C_N(G)$ and $C_S(G)$. We also obtain formulas relating the independence number and the matchin…
The core vertex set of a graph is an invariant of the graph. It consists of those vertices associated with the non-zero entries of the nullspace vectors of a $\{0,1\}$-adjacency matrix. The remaining vertices of the graph form the core--forbidden vertex set. For graphs with independent core vertices, such as bipartite minimal configurations and trees, the nullspace induces a well defined three part vertex partition. The parts of this partition are the core vertex set, their neighbours and the remote core--forbidden vertices. The set of the remote core--forbidden vertices are those not adjacent…
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 $r+s$, …
Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random matrix in ${\mathbb F}_p$, \begin{equation} {\mathbb P}(\text{$A$ is singular in ${\mathbb F}_p$})\leq \frac{1+{\mathrm{o}}(1)}{p-1}, \end{equation} as $n$ goes to infinity. As a consequence, as a random matrix in $\mathbb R$, \begin{equation} {\mathbb P}(\text{$A$ is singular in $\mathbb R$})={\mathrm{o}}(1) \end{equation} as $n$ goes to infinity. This answers an open problem by Frieze [12] and Vu [29…
We complete the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$. The unsigned graphs and the disconnected, the bipartite and the complete signed graphs with this property have already been determined in two earlier papers. Here we deal with the remaining cases.
We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions.
We completely determine the spectrum of an $I$-graph, that is, the eigenvalues of its adjacency matrix. We apply our result to prove known characterizations of connectedness and bipartiteness in $I$-graphs by using an spectral approach. With our result, we also determine the nullity of a certain subfamily of $I$-graphs.
Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,σ)$ is a weighted graph with a special weight function $σ: E(G)\to \{-1,1\}$. A graph is sign-invertible (or sign-invertible) if its inverse is a signed graph. A sign-invertible graph is always unimodular. The inverses of graphs have interesting combinatorial interests. In this paper, we study inverses of graphs and provide a combinatorial description for sign-invertible graphs, whic…
A spectral condition for spanning trees with restricted degrees in bipartite graphs
School of Science, Jiangsu University of Science and Technology（江苏科技大学理学院）；School of Mathematics and Information Science, Yantai University（烟台大学数学与信息科学学院）
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.
The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of $C_6$ with a leaf of $P_{n-6}$\,. Caporossi et al. conjecture that the unicyclic graph with maximal energy is $P_n^6$ for $n=8,12,14$ and $n\geq 16$. In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that $E(P_n^6)$ is maximal within the class of the unicyclic biparti…
