A factor of a graph is a spanning subgraph satisfying some given conditions. An earlier survey of factors can be traced back to the Akiyama and Kano [J. Graph Theory, 1985, 9: 1-42] in which they described the characterization of factors in (bipartite) graphs and digraphs, respectively. Soon after, Kouider and Vestergaard summarized the findings related to connected factors [Graphs Combin., 2005, 21(1): 1-26]. Plummer extended the aforementioned research by providing a comprehensive overview of progress made in the study of graph factors and factorization from 1985 to 2003 [Discrete Math., 2007, 7-8: 791-821]. In this paper, we aim to summarize the relevant results regarding factors from the perspective of eigenvalues.

For a graph G and for two distinct vertices u and v , let kappa (u, v)) be the maximum number of vertex-disjoint paths joining u and v in G . The average connectivity matrix of an n-vertex connected graph G , written A((kappa) over bar)(G), is an n x n matrix whose (u, v))-entry is kappa(u, v)/ ((n)(2)) and let rho (A((kappa) over bar) (G)) be the spectral radius of A((kappa) over bar)(G) . In this paper, we investigate some spectral properties of the matrix. In particular, we prove that for any n-vertex connected graph G , we have rho(A((kappa) over bar)(G)) <= 4 alpha'(G)/n , which implies a result of Kim and O [8] stating that for any connected graph G , we have (kappa) over bar (G) <= 2 alpha'(G), where (kappa) over bar (G) = Sigma(u,v is an element of V (G)) kappa(u, v)G)/((n)(2)) and alpha'(G) is the maximum size of a matching in G ; equality holds only when G is a complete graph with an odd number of vertices. Also, for bipartite graphs, we improve the bound, namely rho(A((kappa) over bar)(G)) <= (n-alpha'(G))(4 alpha'(G)-2)/n (n - 1), and equality in the bound holds only when G is a complete balanced bipartite graph. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We prove upper bounds for the spectral radius ρ(G) of an n-vertex graph with given maximum degree and girth at least 2ℓ+1. This extends the previous result of [13] regarding graphs with girth at least five. When ℓ=3 or |V(G)| is relatively small compared with the maximum degree, our upper bounds are sharp. In addition, for a tree T, we provide an upper bound for the spectral radius of an n-vertex T-free graph with given maximum degree. This bound is also sharp for a certain class of trees.

The matching number of G, written alpha'(G), is the size of a maximum matching in G. Suppose that n and k are positive integers of the same parity. Let theta'(n, k) be the largest root of x(3) - (n - k - 2)x(2) - (n - 1)x + k(n - k - 2) = 0 and theta(n, k) = {theta'(n, k) if n >= 3k +2 n - k -2 + root(n - k - 2)(2) + 4(n(2) - k(2)) if n <= 3k In this article, we prove that for a positive integer n >= k + 2, if G is an n-vertex connected graph with the spectral radius rho(G) > theta(n, k), then alpha'(G) > n-k/2. The bound is sharp in the sense that for every positive integer n >= k + 2, there are graphs H with rho(H) = theta(n, k) and alpha'(G) = n-k/2.

Let G be a graph and let g,f be nonnegative integer-valued functions defined on V(G) such that g(v)≤f(v) and g(v)≡f(v)(mod2) for all v∈V(G). A (g,f)-parity factor of G is a spanning subgraph H such that for each vertex v∈V(G), g(v)≤dH(v)≤f(v) and f(v)≡dH(v)(mod2). We prove sharp upper bounds for certain eigenvalues in an h-edge-connected graph G with given minimum degree to guarantee the existence of a (g,f)-parity factor.

Let $\lambda_2(G)$ and $\kappa'(G)$ be the second largest eigenvalue and the edge-connectivity of a graph $G$, respectively. Let $d$ be a positive integer at least 3. For $t=1$ or 2, Cioaba proved sharp upper bounds for $\lambda_2(G)$ in a $d$-regular simple graph $G$ to guarantee that $\kappa'(G) \ge t+1$. In this paper, we settle down for all $t \ge 3$.

For a graph G , let alpha(G) be the independence number of G , let L(G) be the Laplacian matrix of G , and let mGI be the number of eigenvalues of L(G) in the interval I. Ahanjideh, Akbari, Fakharan and Trevisan proved that alpha(G) <= mG[0, n - alpha(G)] if G is an n-vertex connected graph. Choi, Moon and Park characterized graphs with alpha(G) = mG[0, n - alpha(G)] for alpha(G) = 2 and alpha (G) = n - 2 . In this paper, we give a characterization for alpha (G) = 3 and alpha (G) = n - 3 .(c) 2023 Elsevier Inc. All rights reserved.

For given graphs G and H, the graph G is H-saturated if G does not contain H as a subgraph but for any e∈E(G‾), G+e contains H. In this note, we prove that if G is an n-vertex Kr+1-saturated graph such that for each vertex v∈V(G),∑w∈N(v)dG(w)≥(r−2)d(v)+(r−1)(n−r+1), then ρ(G)≥ρ(Sn,r), where Sn,r is the graph obtained from a copy of Kr−1 with vertex set S by adding n−r+1 vertices, each of which has neighborhood S. This provides a sharp lower bound for the spectral radius in an n-vertex Kr+1-saturated graph for r=2,3, verifying a special case of a conjecture by Kim, Kim, Kostochka and O.

Let $a$ and $b$ be positive integers. An even $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $d_H(v)$ is even and $a \le d_H(v) \le b$. Matsuda conjectured that if $G$ is an $n$-vertex 2-edge-connected graph such that $n \ge 2a+b+\frac{a^2-3a}b - 2$, $\delta(G) \ge a$, and $\sigma_2(G) \ge \frac{2an}{a+b}$, then $G$ has an even $[a,b]$-factor. In this paper, we provide counterexamples, which are highly connected. Furthermore, we give sharp sufficient conditions for a graph to have an even $[a,b]$-factor. For even $an$, we conjecture a lower bound for $\lambda_1(G)$ in an $n$-vertex graph to have an $[a,b]$-factor, where $\lambda_1(G)$ is the largest eigenvalue of $G$.

In this paper, we prove that for a digraph D, we have rho(D) <= root max(v is an element of V (D)) sigma(-)(u is an element of N)(v) d(+)(u), where for a vertex v is an element of V(D), d(+)(v) is the number of vertices u such that vu is an arc. As a result, we prove chi(D) <= 1 + root max(v is an element of V (D)) sigma(-)(u is an element of N)(v) d(+)(u). We also prove that rho(D) <= root m - delta(+)(D)(1 + n(min)'(D)), where m is the number of arcs in D, delta(+)(D) = min(v is an element of V (D)) d(+)(v), and n(min)'(D) = min(v is an element of V (D)) |{u : uv is an arc but vu is not an arc}|. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

In 1972, Cvetković proved that if G is an n-vertex simple graph with the chromatic number k, then its spectral radius is at most the spectral radius of the n-vertex balanced complete k-partite graph. In this paper, we analyze the characteristic polynomial of a digraph D to prove a tight upper bound for the spectral radius of D in terms of the number of vertices and the chromatic number of D; we also characterize when equality holds. This provides a simple proof of a result by Lin and Shu [6].

For positive integers, r ≥ 3 , h ≥ 1 , and k ≥ 1 , Bollobás, Saito, and Wormald proved some sufficient conditions for an h ‐edge‐connected r ‐regular graph to have a k ‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected r ‐regular graph to have a k ‐factor in 2010. Gu found an upper bound for certain eigenvalues in an h ‐edge‐connected r ‐regular graph to have a k ‐factor in 2014. For positive integers a ≤ b , an even (or odd) [ a , b ] ‐factor of a graph G is a spanning subgraph H such that for each vertex v ∈ V ( G ) , d H ( v ) is even (or odd) and a ≤ d H ( v ) ≤ b . In this paper, we prove upper bounds (in terms of a , b , and r ) for certain eigenvalues (in terms of a , b , r , and h ) in an h ‐edge‐connected r ‐regular graph G to guarantee the existence of an even [ a , b ] ‐factor or an odd [ a , b ] ‐factor. This result extends the one of Bollbás, Saito, and Wormald, the one of Lu, and the one of Gu.

Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $g(v) \le d_H(v) \le f(v)$ and $f(v)\equiv d_H(v) \pmod{2}$. We prove sharp upper bounds for certain eigenvalues in an $h$-edge-connected graph $G$ with given minimum degree to guarantee the existence of a $(g,f)$-parity factor; we provide graphs showing that the bounds are optimal. This result extends the recent one of the second author (2022), extending the one of Gu (2014), Lu (2010), Bollb{\'a}s, Saito, and Wormald (1985), and Gallai (1950).

For a graph H, a graph G is H-saturated if G does not contain H as a subgraph but for any e∈E(G¯), G+e contains H. In this note, we prove a sharp lower bound for the number of paths and walks on length 2 in n-vertex Kr+1-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed r and n→∞.

A perfect matching in a graph G is a set of disjoint edges covering all vertices of G. Let ρ(G) be the spectral radius of a graph G, and let θ(n) be the largest root of x3−(n−4)x2−(n−1)x+2(n−4)=0. In this paper, we prove that for a positive even integer n≥8 or n=4, if G is an n-vertex graph with ρ(G)>θ(n), then G has a perfect matching; for n=6, if ρ(G)>1+332, then G has a perfect matching. It is sharp for every positive even integer n≥4 in the sense that there are graphs H with ρ(H)=θ′(n) and no perfect matching, where θ′(n)=θ(n) if n=4 or n≥8 and θ′(6)=1+332.

An odd [1, b]-factor of a graph G is a spanning subgraph H such that for each vertex v is an element of V(G), d(H)(v) is odd and 1 <= d(H)(v) <= b. Let lambda(3)(G) be the third largest eigenvalue of the adjacency matrix of G. For positive integers r >= 3 and even n, Lu et al. (2010) proved a lower bound for lambda(3)(G) in an n-vertex r-regular graph G to guarantee the existence of an odd [1, b]-factor in G. In this paper, we improve the bound; it is sharp for every r. (C) 2020 Elsevier B.V. All rights reserved.

The k-independence number of a graph G is the maximum size of a set of vertices at pairwise distance greater than k. In this paper, for each positive integer k, we prove sharp upper bounds for the k-independence number in an n-vertex connected graph with given minimum and maximum degree.

For a graph H, a graph G is H-saturated if G does not contain H as a subgraph but for any e ∈ E(G), G+e contains H. In this note, we prove a sharp lower bound for the number of paths and walks on length 2 in n-vertex K_r+1-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed r and n→∞.

Let mu(2)(G) be the second smallest Laplacian eigenvalue of a graph G. The vertexconnectivity of G, written kappa(G), is the minimum size of a vertex set S such that G- S is disconnected. Fiedler proved that mu(2)(G) <= kappa(G) for a non-complete simple graph G; for this reason mu(2)(G) is called the "algebraic connectivity'' of G. His result can be extended to multigraphs: for any multigraph G who underlying graph is not a complete graph, we have mu(2)(G) <= kappa (G)m(G), where for a pair of vertices u and v, let m(u, v) be the number of edges with endpoints u and v and m(G) = max((u,v)epsilon E(G)) m(v, u). Let lambda(2)(G) be the second largest eigenvalue of a graph G. We also prove that for any dregular multigraph G whose underlying graph is not the complete graph with 2 vertices, if lambda(2)(G) < 3/4 d, then G is 2-connected. For t >= 2 and infinitely many d, we construct dregular multigraphs H with lambda(2)(H) = 0, epsilon(H) = t, and m(H) = d/t. These graphs show that the inequality mu(2)(G) = kappa(G)m(G) is sharp and that there is no upper bound for.2(G) in a d-regular multigraph G to guarantee a certain vertex-connectivity greater than or equal to 3. (C) 2019 Elsevier B.V. All rights reserved.

It is proved that for any finite connected graph $G$, there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of $G$ (with equality if and only if $G$ is a tree). This resolves a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs.