The question of which graphs are determined by their spectra is a central problem in spectral graph theory. In this paper, we investigate spectral determination for complement-reducible graphs, or cographs. By combining the Johnson–Newman theorem on generalized cospectrality with standard tools in the asymptotic enumeration of trees, we prove a Schwenk-type result for cographs: almost all cographs have a generalized cospectral mate within the family of cographs, and hence have a cospectral mate. This contrasts sharply with threshold graphs, a well-studied subclass of cographs, where no threshold graph has a cospectral mate within that family. We also establish a signless Laplacian analogue by proving that almost all cographs have a generalized Q-cospectral mate.

Let G=(V,E) be a connected graph. A dominating set is a subset D ⊂ V such that every vertex of G is either in D or adjacent to a vertex in D. A connected dominating set is a dominating set in G which induces a connected subgraph. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). Finding the connected domatic number of general graphs is NP-hard. In this paper, we study the connected domatic number for a well-known family of graphs - the generalized Petersen graphs GP(n, k). Determining the connected domatic number of GP(n, k) is equivalent to determining whether it has two disjoint connected dominating sets. We identify several classes of generalized Petersen graphs with two disjoint connected dominating sets. Moreover, for small k (2 ≤ k ≤ 5), we provide necessary and sufficient conditions for a generalized Petersen graph GP(n, k) to contain two disjoint connected dominating cycles.

The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed by Wang, Wang, and Zhu (European J. Combin., 2023), which asserts that the square-root polynomial of the invariant polynomial Φ_p(G;x) ∈𝔽_p[x] can replace its square-free part to yield a more effective criterion for a graph to be determined by its generalized spectrum (DGS). A key ingredient of our proof is a novel algebraic factorization: we show that the polynomial Φ_p(G;x) is the product of the characteristic polynomials of the adjacency operator restricted to the left null space of the walk matrix and its radical, respectively. Based on this refined DGS-criterion, a broad family of DGS-graphs is constructed via rooted products, significantly generalizing the recent result of Wang, Shen, and Mao (Discrete Appl. Math., 2026).

Godsil and Sun asked whether, for a strongly regular graph X and any two different edges e and f, the edge-deleted graphs X∖ e and X∖ f are degree-similar. We give an affirmative answer to the problem of Godsil and Sun. In fact, we prove the stronger statement that if X is a 1-walk-regular graph, then for any two edges e and f of X, the graphs X∖ e and X∖ f are orthogonally degree-similar. The proof is based on an edge version of the orthogonal-intertwiner method: the equality of the Gram matrices of the projected endpoint vectors in every eigenspace yields an orthogonal matrix commuting with the adjacency matrix and sending one pair of ordered endpoint vectors to the other.

In this paper, we introduce the concepts of positive and negative p-energies of graphs and investigate their behavior under edge addition. Specifically, we generalize the classical notions of positive and negative square energies to the p-energy setting, denoted by E-p(+)(G) and E-p(-)(G), respectively. We establish improved lower bounds for these quantities under edge addition, which sharpen existing results by Abiad et al. in the case p = 2. Furthermore, we address the monotonicity problem for E-p(+) (G) under edge addition, and construct a family of counterexamples showing that monotonicity fails for 1 <= p < 3. Finally, we conclude with several open problems for further investigation. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let n and p denote the numbers of vertices and labels, respectively, in an undirected edge-labeled graph. Previous work showed that, under the Exponential Time Hypothesis (ETH), there is no deterministic algorithm with running time (np)^o(log n/(loglog n)^2). In this paper, we give a deterministic reduction that strengthens this conditional running-time lower bound to (np)^o(log n)poly(|E|). The lower bound holds even for simple edge-labeled graphs. Since our reduction is deterministic, the same lower bound applies to bounded-error randomized algorithms under the randomized Exponential Time Hypothesis. On the resulting hard family, p=n^O(1). Thus, under rETH, the lower bound matches the exponent order of the known randomized quasi-polynomial exact upper bound up to constant factors.

For a given simple graph G, the p-energy of G, denoted by Ep(G), is defined as the sum of the p-th power of the absolute values of the eigenvalues of its adjacency matrix. Let Sn denote the star graph with one internal node and n−1 leaves. Nikiforov conjectured that for 1<p<2, the connected graph of order n with the smallest p-energy is Sn. Recently, this conjecture was proved for bipartite graphs. In this paper, by employing a Coulson-Jacobs-type formula and certain spectral radius results for connected graphs, we completely resolve this conjecture. Furthermore, we establish that the equality condition in the inequality Ep(G)≥Ep(Sn) holds if and only if G is Sn.

For an n-vertex graph G and a rooted graph H(v) with v as the root, the rooted product graph GoH(v) is obtained from G and n copies of H by identifying the root of the ith copy of H with the ith vertex of G for each i. As a refinement of the controllability criterion of Go H(v) obtained recently by Shan and Liu (2025), we obtain an explicit formula for the determinant of the walk matrix of Go H(v). Furthermore, for an important family of graphs .F that are determined by their generalized spectrum (DGS), we introduce the concept of .F-preservers and provide a sufficient condition for a rooted graph to be an .F-preserver. A list of .F-preservers of small order is provided, which leads to many new infinite families of DGS-graphs using rooted products. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let $\Sigma$ be an $n$-vertex controllable or almost controllable signed bipartite graph, and let $\Delta_\Sigma$ denote the discriminant of its characteristic polynomial $\chi(\Sigma; x)$. We prove that if (\rmnum{1}) the integer $2^{ -\lfloor n/2 \rfloor }\sqrt{\Delta _{\Sigma}}$ is squarefree, and (\rmnum{2}) the constant term (even $n$) or linear coefficient (odd $n$) of $\chi(\Sigma; x)$ is $\pm 1$, then $\Sigma$ is determined by its generalized spectrum. This result extends a recent theorem of Ji, Wang, and Zhang [Electron. J. Combin. 32 (2025), \#P2.18], which established a similar criterion for signed trees with irreducible characteristic polynomials.

Spectral characterization of graphs for various graph matrices constitutes a central topic in spectral graph theory. Let G be a graph with adjacency matrix A(G), diagonal degree matrix (G), distance matrix D(G), and transmission matrix (G), respectively. Recently, Alfaro and Zapata (2024) introduced the degree-distance matrices (G)=(G)+D(G) and (G)=(G)-D(G), together with the transmission-adjacency matrices (G)=(G)+A(G) and (G)=(G)-A(G). Based on computational evidence for trees on at most 20 vertices, they conjectured that all trees are determined by the spectra of as well as . In this paper, we disprove these conjectures by constructing an infinite family of pairs of non-isomorphic trees. More precisely, for each integer r≥ 3, we construct a pair of trees on 17r-15 vertices which are simultaneously cospectral with respect to the following six matrices A, L, Q, D, , . The construction is based on an r-regularized leaf extension and an equitable-partition reduction. We also record a simple sign-switching observation for transmission-adjacency matrices: if G is bipartite, then (G) and (G) are similar via a diagonal {±1}-matrix and have the same Smith normal form. Consequently, for trees, the spectral and Smith normal form problems for and are equivalent.

Let G be a simple graph of order n with adjacency matrix A= (a_ij). The determinant and the permanent of the matrix A are defined as detA= ∑_σ∈ S_nsgn(σ) ∏_i=1^n a_iσ(i) and perA= ∑_σ∈ S_n∏_i=1^n a_iσ(i), respectively. The polynomials ϕ(G;x) =det(xI-A(G)) and π(G;x) =per(xI-A(G)) are called the characteristic polynomial and the permanental polynomial of G, respectively. Two graphs are said to be nearly cospectral with respect to the determinant (resp. permanent) if the difference of their characteristic (resp. permanental) polynomials is a constant. Lv et al. introduced the nearly cospectral graphs problem with respect to the determinant, and provided partial results in the case modulo 4. In this paper, we mainly prove that the corresponding results also hold for the nearly cospectral graphs problem with respect to the permanent. The determinant and permanent are the immanants corresponding to the irreducible characters (1^n) and (n) of the symmetric group S_n, respectively. Here, the immanant d_λ(A) of A is defined as d_λ(A) = ∑_σ∈ S_n χ_λ(σ) ∏_i=1^n a_iσ(i), where χ_λ is the irreducible character of S_n indexed by the partition λ. The immanantal polynomial of G associated with χ_λ is given by ϕ_λ(G;x)=d_λ(xI-A). In this paper, we also establish a similar result for nearly immanantal cospectral graphs in 𝔽_2[x] for all irreducible characters χ_λ.

Characterizing graphs uniquely determined by their spectra (DS) is a core open problem in spectral graph theory. While this problem has been extensively investigated for simple undirected graphs, it remains relatively underexplored for oriented graphs. For a simple undirected graph G equipped with an orientation σ, the corresponding oriented graph Σ=(G,σ) is the digraph obtained by orienting each edge of G according to σ. An oriented graph Σ is said to be determined by its generalized skew spectrum (DGSS) if every oriented graph sharing the same generalized skew spectrum is isomorphic to Σ. This paper develops a new sufficient criterion for recognizing DGSS controllable oriented graphs, which applies to a much broader family of graphs than previously known results. Let S be the skew-adjacency matrix of Σ, W(Σ)=[e,Se,…,S^n-1e], and d_n the last invariant factor of W(Σ). For each odd prime p, we define the polynomial Φ_p(Σ;x)=(χ(S;x),χ(S+J;x)) over the finite field 𝔽_p, which is invariant under generalized skew cospectrality. By analyzing the square-free part of Φ_p(Σ;x) and the associated p-main polynomial, we establish a DGSS sufficient condition under the square-free assumption on d_n. The proposed criterion allows higher p-nullity and recovers the square-free determinant criterion of Qiu, Wang and Wang (2019) as a special case. We further provide illustrative examples to verify the wider applicability of our new condition and to highlight the role of the compatibility constraints on the irreducible factors of Φ_p(Σ;x).

The spectral characterization of graphs is a central theme in spectral graph theory. A graph G is determined by its spectrum (DS) if every graph cospectral with G is also isomorphic to G. The definition is extended to the generalized spectrum, where a graph G is determined by its generalized spectrum (DGS) if any graph H that is cospectral with G and whose complement is cospectral with (G)over bar must be isomorphic to G. While it is clear that all DS graphs are also DGS, the reverse is not always true. This leads to a natural, unanswered question: Which graphs are DGS but not DS? Previous research has focused on identifying graphs that are either DS or DGS, but, to our knowledge, research on this specific problem has not attracted much attention. This paper addresses the problem by introducing an infinite family of graphs that are DGS but not DS. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We present a determinantal formula for the number of spanning trees of a complete multipartite graph containing a given spanning forest. Our approach relies on the Generalized Matrix Determinant Lemma and Jacobi's formula for the derivative of a determinant. This work generalizes known results for complete bipartite graphs and offers an algebraic perspective on the problem. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Point cloud video representation learning is crucial for 3D dynamic scene understanding. In this paper, we propose MoSaiC, a novel Motion-Saliency Complementary masked modeling framework for self-supervised point cloud video representation learning. MoSaiC couples three components: Curriculum Motion-Saliency Masking (CMSM), which guides the masking process toward motion-salient tokens under a curriculum schedule; Normal-Flow Motion (NFM) modeling, which supervises the local rigid rotation of each token in the Lie algebra so(3) as an explicit geometric motion target; and Cross-view Token Consistency Prediction (CTCP), which enforces consistency between two complementary masked views at the token level. Together, these components allow MoSaiC to effectively capture both appearance and motion dynamics. Extensive experiments on multiple downstream tasks, including action recognition, temporal action segmentation, and point-level semantic segmentation, demonstrate the effectiveness of our approach.

Orthogonal time frequency space (OTFS)/vector OFDM (VOFDM) is widely regarded as a promising waveform for next-generation mobile communications. However, its spectral characteristics are not yet fully understood. The bandwidth allocation scheme, which is crucial for OTFS's integration into practical wireless standards, also remains unexplored. In this paper, we investigate the spectral characteristics of OTFS signals by analyzing their power spectral density (PSD). We demonstrate that the PSD of discrete-time OTFS signals is periodic with a period of M1/T-s, where M is the size of the time/Doppler domain in OTFS, a.k.a., the vector size in VOFDM, and T-s is the sampling interval length of digital to analog converter (DAC), resulting in M identical spectral components within the spectral range [- 1/2T(s), 1/2T(s)) of the continuous-time OTFS signal. The periodicity makes bandwidth allocation for OTFS/VOFDM signaling substantially challenging. Furthermore, we establish a relationship between the PSD of OFDM signals and that of OTFS signals, revealing that, when the information symbols are independent, the PSD of OTFS signals is equal to the sum of the PSDs of the component-expanded OFDM (CEP-OFDM) signals. Lastly, we derive a relationship between the information symbols and the corresponding OTFS spectrum, and based on which, we propose a null-space-based linear precoding (NSLP) method for OTFS signals to enable flexible bandwidth allocation. Numerical results validate our analytical results regarding the PSD of OTFS signals and show the effectiveness of our proposed NSLP method in tailoring the spectrum of OTFS signals.

We provide a criterion to distinguish two graphs which are indistinguishable by 2-dimensional Weisfeiler-Lehman algorithm for almost all graphs. Haemers conjectured that almost all graphs are identified by their spectrum. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectrum.

Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix. The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is $[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circ P_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of $P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signless Laplacian counterpart of the following recently established identity [17]: $$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and $W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unify the above two equalities.

Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_\mu(G):=A(G)-\mu D(G)$. Two graphs $G_1$ and $G_2$ are called degree-similar if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M =A(G_2)$ and $M^{-1} D(G_1) M =D(G_2)$. In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees $G_1$ and $G_2$ such that $tI- L_\mu(G_1)$ and $tI-L_\mu(G_2)$ have the same Smith normal form over ${\mathbb{Q}}(\mu)[t]$, which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph $G$ and any two edges $e$ and $f$ of $G$, $G \backslash e$ and $G \backslash f$ have identical $\mu$-polynomial, i.e., $\det(tI-L_\mu(G \backslash e))=\det(tI-L_\mu(G \backslash f))$, which enables the construction of pairs of non-isomorphic graphs with same $\mu$-polynomial, where $G \backslash e$ denotes the graph obtained from $G$ by deleting the edge $e$.

Over a decade ago, Koolen, Hayat, and Iqbal posed the problem of whether the distance spectrum determines bipartiteness within the class of connected graphs. In this paper, we resolve this problem in the negative: we explicitly construct an infinite family of counterexamples, where each pair comprises a connected bipartite graph and a connected non-bipartite graph with equal distance spectra.