Identifying and constructing graphs that are determined by their generalized spectrum (DGS) is a significant and challenging problem in spectral graph theory. Recently, a simple criterion for almost controllable graphs to be DGS was proposed by Lin et al. (2026), utilizing the modified walk matrix. In this paper, we investigate the evolution of the modified walk matrix under disjoint union and join operations with a singleton vertex. We establish an exact algebraic identity for the determinant of the modified walk matrix of the resulting graph. Based on this identity and the DGS-criterion of Lin et al., we construct infinite families of almost controllable graphs that are DGS, extending the previous construction of Liu et al. (2019), which was restricted to controllable graphs.
A graph G is called a chain graph if it contains none of the graphs {2K2,C3,C5} as induced subgraphs. It is said to be almost controllable if it has exactly one non-main eigenvalue, that is, precisely one eigenvalue whose corresponding eigenspace is orthogonal to the all-ones vector. Within the class of chain graphs, w...
B. Alshamary, M. Andelic· Malaysian journal of mathema...· 0 citations
All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2^{-\Omega(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integr...
T. Manna, Supriyo Dutta, Baby Bhattacharya· 0 citations
In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters d1≤d2, there exists a coupling of G(n,d1) and G(n,d2) satisfying that G(n,d1)⊆G(n,d2) with high probability, confirming a conjecture of Ga...
Lawrence Hollom, Lyuben Lichev, Adva Mond et al.· The Annals of Applied Probab...· 0 citations
In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J:...
Qing-Cheng Zeng, Cheng Zeng, Yu-Mei Xue et al.· 0 citations
A Neumaier graph is a non-complete edge-regular graph containing a regular clique; it is called strictly Neumaier if it is not strongly regular. In this paper we present a construction using finite rings that unifies several known results and yields three new families, each containing infinitely many strictly Neumaier...
A. Abiad, W. Castryck, M. De Boeck et al.· 0 citations
The notion of graph complements has been widely generalized to study diverse structural and spectral properties of graphs. In this paper, we introduce and investigate the concept of generalized color complements of graphs with respect to a prescribed vertex partition. Building on earlier work on generalized color compl...
S. Sahana, S. D'Souza, S. Nayak et al.· Discrete Mathematics, Algori...· 0 citations
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