Tutte polynomial of a small-world farey graph

In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of spanning trees, as well as the number of connected spanning subgraphs. Furthermore, we also derive exact expressions for the chromatic polynomial and the reliability polynomial of these graphs.Comment: 6 page

Square nearly nonpositive sign pattern matrices

AbstractA sign pattern matrix A is called square nearly nonpositive if all entries but one of A2 are nonpositive. We characterize the irreducible sign pattern matrices that are square nearly nonpositive. Further we determine the maximum (resp. minimum) number of negative entries that can occur in A2 when A is irreducible square nearly nonpositive (SNNP), and then we characterize the sign patterns that achieve this maximum (resp. minimum) number. Finally, we discuss some spectral properties of the sign patterns which are square nonpositive or square nearly nonpositive

Maxima of the index: forbidden unbalanced cycles

This paper aims to address the problem: what is the maximum index among all $\mathcal{C}^-_r$-free unbalanced signed graphs, where $\mathcal{C}^-_r$ is the set of unbalanced cycle of length $r$. Let $\Gamma_1 = C_3^- \bullet K_{n-2}$ be a signed graph obtained by identifying a vertex of $K_{n-2}$ with a vertex of $C_3^-$ whose two incident edges in $C_3^-$ are all positive, where $C_3^-$ is an unbalanced triangle with one negative edge. It is shown that if $\Gamma$ is an unbalanced signed graph of order $n$, $r$ is an integer in $\{4, \cdots, \lfloor \frac{n}{3}\rfloor + 1 \}$, and $$\lambda_{1}(\Gamma) \geq \lambda_{1}(\Gamma_1),$$ then $\Gamma$ contains an unbalanced cycle of length $r$, unless $\Gamma \sim \Gamma_1$. \indent It is shown that the result are significant in spectral extremal graph problems. Because they can be regarded as a extension of the spectral Tur{\'a}n problem for cycles [Linear Algebra Appl. 428 (2008) 1492--1498] in the context of signed graphs. Furthermore, our result partly resolved a recent open problem raised by Lin and Wang [arXiv preprint arXiv:2309.04101 (2023)].Comment: 13pages,5figure

On the eigenvalues and Seidel eigenvalues of chain graphs

In this paper we consider the eigenvalues and the Seidel eigenvalues of a chain graph. An\dbareli\'{c}, da Fonseca, Simi\'{c}, and Du \cite{andelic2020tridiagonal} conjectured that there do not exist non-isomorphic cospectral chain graphs with respect to the adjacency spectrum. Here we disprove this conjecture. Furthermore, by considering the relation between the Seidel matrix and the adjacency matrix of a graph, we solve two problems on the number of distinct Seidel eigenvalues of a chain graph, which was posed by Mandal, Mehatari, and Das \cite{mandal2022spectrum}.Comment: 13 pages, 2 figure

Bounds on the largest eigenvalues of trees with a given size of matching

AbstractVery little is known about upper bound for the largest eigenvalue of a tree with a given size of matching. In this paper, we find some upper bounds for the largest eigenvalue of a tree in terms of the number of vertices and the size of matchings, which improve some known results

Extremal results for Kr+1−\mathcal{K}^-_{r + 1}-free signed graphs

This paper gives tight upper bounds on the number of edges and the index for $\mathcal{K}^-_{r + 1}$-free unbalanced signed graphs, where $\mathcal{K}^-_{r + 1}$ is the set of $r+1$-vertices unbalanced signed complete graphs. \indent We first prove that if $\Gamma$ is an $n$-vertices $\mathcal{K}^-_{r + 1}$-free unbalanced signed graph, then the number of edges of $\Gamma$ is $$e(\Gamma) \leq \frac{n(n-1)}{2} - (n - r ).$$ \indent Let $\Gamma_{1,r-2}$ be a signed graph obtained by adding one negative edge and $r - 2$ positive edges between a vertex and an all positive signed complete graph $K_{n - 1}$. Secondly, we show that if $\Gamma$ is an $n$-vertices $\mathcal{K}^-_{r + 1}$-free unbalanced signed graph, then the index of $\Gamma$ is $$\lambda_{1}(\Gamma) \leq \lambda_{1}(\Gamma_{1,r-2}),$$ with equality holding if and only if $\Gamma$ is switching equivalent to $\Gamma_{1,r-2}$. \indent It is shown that these results are significant in extremal graph theory. Because they can be regarded as extensions of Tur{\'a}n's Theorem [Math. Fiz. Lapok 48 (1941) 436--452] and spectral Tur{\'a}n problem [Linear Algebra Appl. 428 (2008) 1492--1498] on signed graphs, respectively. Furthermore, the second result partly resolves a recent open problem raised by Wang [arXiv preprint arXiv:2309.15434 (2023)].Comment: 13 pages, 1 figur

A CLASS OF UNICYCLIC GRAPHS DETERMINED BY THEIR LAPLACIAN SPECTRUM

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p+4