For a group Ω, the associated power graph P(Ω) is defined as the graph whose vertices are the elements of Ω, with two distinct vertices u,v∈ Ω being adjacent if either u=v^m or v=u^n for some m,n ∈ℕ. In this paper, we completely characterise the structure of the power graph associated with the class of metacyclic groups. Building on this structural description, we derive explicit expressions for the characteristic polynomials of the adjacency, Laplacian, and signless Laplacian matrices. Moreover, we obtain lower and upper bounds for the spectral radii of the adjacency and signless Laplacian matrices.

The generalized reciprocal distance matrix of a graph 𝒢, denoted by RD_α(𝒢), is defined as RD_α(𝒢)=α RT_r(𝒢)+(1-α) RD(𝒢), α∈[0,1], where RT_r(𝒢) represents the diagonal matrix of reciprocal vertex transmissions, and RD(𝒢) is the Harary (reciprocal distance) matrix of 𝒢. In this paper, we investigate the RD_α-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of RD_α(𝒢) when 𝒢 is formed as a joined union of regular graphs. These results provide closed-form expressions for the corresponding spectra of several important graph classes. Moreover, we show that the power graphs of the dihedral group D_2n and the generalized quaternion group Q_4n admit representations as joined union graphs. Using this structural characterization, we determine the RD_α-spectra of power graphs arising from various classes of finite groups, including cyclic groups ℤ_n, dihedral groups D_2n, generalized quaternion groups Q_4n, elementary abelian p-groups, and certain non-abelian groups of order pq.

For a finite group Ω , the power graph P(Ω ) is a simple connected graph where the set of vertices consists of the elements of the group Ω and two vertices in P(Ω ) are adjacent if and only if one is an integral power of the other. In this paper, we explore the distance spectrum and distance signless Laplacian spectrum of the power graph over a class of split metacyclic groups. We provide lower and upper bounds for the distance spectral radius of the power graph for both the split metacyclic group and the finite cyclic group. Additionally, we establish bounds for the distance signless Laplacian spectral radius of the power graph of a split metacyclic group.

The power graph P(G) of a finite group G is the graph with vertex set G and edge set E(P(G))={uv: u,v ∈ G, u ≠ v, u ∈⟨ v ⟩ or v ∈⟨ u ⟩}, where ⟨ x⟩ denotes the cyclic subgroup generated by x. In this paper, we characterise all the finite groups with exponent q whose power graphs are friendship graphs, firefly-type graphs, or torch graphs. We prove that the power graph of a finite group G with exponent q is a friendship graph if and only if q=3. In particular, in the abelian case, this is equivalent to G≅ℤ_3^n. We further show that, among all the symmetric and alternating groups, only S_3 and A_4 have firefly-type power graphs, whereas no finite group has a power graph isomorphic to a torch graph. Finally, we determine the generalised distance spectra D_α-spectra of these graph classes.

There are 22 types of doubly semi-equivelar maps, with curvature 0, on the plane which provide infinitely many doubly semi-equivelar maps of respective types on the torus. In this article, we show that every such doubly semi-equivelar map on the torus contains a Hamiltonian cycle. As a consequence, we establish the Nash-Williams conjecture for the graphs associated with these doubly semi-equivelar maps by showing that these graphs are either 3-connected or 4-connected.

The A_α matrix of a graph G is defined as A_α(G) = α D(G) + (1-α )A(G) , where D(G) and A(G) denote the degree diagonal matrix and adjacency matrix of the graph G, respectively. In this article, we determine the eigenvalues of A_α matrix for the power graph of a class of metacyclic groups. We set upper and lower bounds for the largest eigenvalues of A_α matrix associated with the power graphs of the finite cyclic group of order n and the metacyclic group.

The power graph P(G) of a group G is a simple graph with the vertex set G such that two distinct vertices u,v is an element of G are adjacent in P(G) if and only if u(m) = v or v(m) = u, for some m is an element of N. The purpose of this paper is to introduce the notion of the power graph for a gyrogroup. Using this, we investigate the combinatorial properties of power graph of a gyrogroup, G(k), of order 2(k) for k >= 3. In particular, we determine the Hamiltonicity and planarity of the power graph of G(k). Consequently, we calculate distance properties, resolving polynomial, independent domination polynomial, Hosoya and reciprocal Hosoya polynomials, characteristic polynomials, and the spectral radius of P(G(k)).

Semi-Equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the 2-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices.

A vertex v in a map M has the face-sequence (p_1^n_1. p_2^n_2. … . p_k^n_k) , if consecutive n_i numbers of p_i -gons are incident at v in the given cyclic order for 1 ≤ i ≤ k . A map is called semi-equivelar if the face-sequence of each vertex is same throughout the map. A doubly semi-equivelar map is a generalization of semi-equivelar map which has precisely 2 distinct face-sequences. In this article, we determine all the types of doubly semi-equivelar maps of combinatorial curvature 0 on the Klein bottle. We present classification of doubly semi-equivelar maps on the Klein bottle and illustrate this classification for those doubly semi-equivelar maps which comprise of face-sequence pairs {(3^6), (3^3.4^2)} and {(3^3.4^2), (4^4)} .

The power graph denoted by 𝒫(𝒢) of a finite group 𝒢 is a graph with vertex set 𝒢 and there is an edge between two distinct elements u, v ∈𝒢 if and only if u^m = v or v^m = u for some m ∈ℕ. Depending on the distance, the Hosoya polynomial contains a lot of knowledge about graph invariants which can be used to determine well-known chemical descriptors. The Hosoya index of a graph Γ is the total number of matchings in Γ. In this article, the Hosoya properties of the power graphs associated with a finite group, including the Hosoya index, Hosoya polynomial, and its reciprocal are calculated.

The power graph G = P(Ω) of a finite group Ω is a graph with the vertex set Ω and two vertices u, v ∈Ω form an edge if and only if one is an integral power of the other. Let D(G), A(G), RT(G), and RD(G) denote the degree diagonal matrix, adjacency matrix, the diagonal matrix of the vertex reciprocal transmission, and Harary matrix of the power graph G respectively. Then the A_α and RD_α matrices of G are defined as A_α(G) = α D(G) + (1-α)A(G) and RD_α(G) = α RT(G) + (1-α)RD(G). In this article, we determine the eigenvalues of A_α and RD_α matrices of the power graph of group 𝒢 = ⟨ s,r : r^2^kp = s^2 = e, srs^-1 = r^2^k-1p-1⟩. In addition, we calculate its distant and detotar distance degree sequences, metric dimension, and strong metric dimension.

A map is called 2-semi equivelar if it has exactly two distinct cyclic arrangement of faces at its vertices. A 2-semi-equivelar map is called 2-uniform if it has precisely 2 orbits of vertices under its symmetric group. Doubly semi-equivelar maps are a subclass of 2-semi equivelar maps that are used to determine 2-uniform maps. In this article, we determine doubly semi-equivelar maps of curvature 0 on the plane and torus exhaustively. Further, we present a classification of doubly semi-equivelar maps on the torus and illustrate this for those doubly semi-equivelar maps which comprise face-sequence pairs {(3(6)),(3(3).4(2))} and {(3(3).4(2)),(4(4))}.

The power graph P(G) of a group G is a simple graph with the vertex set G such that two distinct vertices u,v ∈ G are adjacent in P(G) if and only if u^m = v or v^m = u, for some m ∈ℕ. The purpose of this paper is to introduce the notion of a power graph for gyrogroups. Using this, we investigate the combinatorial properties of a certain gyrogroup, say G(n), of order 2^n for n ≥ 3. In particular, we determine the Hamiltonicity and planarity of the power graph of G(n). Consequently, we calculate distant properties, resolving polynomial, Hosoya and reciprocal Hosoya polynomials, characteristic polynomials, and the spectral radius of the power graph of G(n).

The power graph P(Ω) of a group Ω is a graph with the vertex set Ω such that two distinct vertices form an edge if and only if one of them is an integral power of the other. In this article, we determine the power graph of the group 𝒢 = ⟨ s,r : r^2^kp = s^2 = e, srs^-1 = r^2^k-1p-1⟩. Further, we compute its characteristic polynomial for the adjacency, Laplacian, and signless Laplacian matrices associated with this power graph. In addition, we determine its spectrum, Laplacian spectrum, and Laplacian energy.

The well-known twenty types of 2-uniform tilings of the plane give rise infinitely many doubly semi-equivelar maps on the torus. In this article, we show that every such doubly semi-equivelar map on the torus contains a Hamiltonian cycle. As a consequence, we establish the Nash-Williams conjecture for the graphs associated with these doubly semi-equivelar maps by showing that these graphs are either 3-connected or 4-connected.

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.