Let \(L(G(k))\) be a line graph of a \(k\)-subdivided graph \(G(k)\) of any connected, simple and undirected graph \(G\). In this paper, we fixed the valency dependence invariants of \(L(G(k))\) for \(k\geq2.\) Our results are the generalization of the results proved in [1,3,8].

A star edge coloring of a graph \(G\) is a proper edge coloring in which every path or cycle on four vertices contains at least three distinct colors. The star edge chromatic number of \(G\), denoted by \(\chi^\prime_{\mathrm{st}}(G)\), is the minimum number of colors required for such a coloring. In this paper, we study the star edge chromatic number of several graph classes derived from the Knödel graph \(W_{\Delta,n}\). In particular, we determine bounds and exact values for \(\chi^\prime_{\mathrm{st}}(G)\) for the Knödel graph \(W_{\Delta,n}\), its middle graph \(M\!\left(W_{\Delta,n}\right)\), and its Mycielskian \(\mu\!\left(W_{\Delta,n}\right)\). Furthermore, we provide explicit constructions of star edge colorings that attain these values.

Let \(G\) be a simple connected mixed graph, and let \(H(G)\) denote the Hermitian adjacency matrix of \(G\). The Hermitian permanental polynomial of \(G\) is defined as \(\pi(G; x) = \operatorname{per}(xI – H(G))\), where \(\operatorname{per}(\cdot)\) represents the permanent and \(I\) is the identity matrix. In this paper, we first derive fundamental properties of the Hermitian permanental polynomial for mixed graphs and establish explicit formulas relating its coefficients to those of the characteristic polynomial. We then analyze the root distribution of this polynomial, determining the number of zero roots for several special classes of mixed graphs. Finally, we characterize mixed graphs that remain cospectral under four‑way switching and prove that this operation preserves the permanental spectrum.

The Euler Sombor (\(EU\)) index of a graph \(G\) is defined as \[EU(\mathit{G})=\sum \limits_{{\mathit{u}}{\mathit{v}}\in E(\mathit{G})} \sqrt{deg_G^2(u)+deg_G^2(v)+deg_G(u)deg_G(v)},\] where \(deg_G(u)\) and \(deg_G(v)\) are the degrees of the vertices \(u\) and \(v\) in the graph \(G\), respectively. Biswaranja Khanra and Shibsankar Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun. Math. Comput. Chem., 94 (2025) 525–548] posed an open problem to determine the extremal values and extremal graphs of the Euler Sombor index in the class of all connected graphs with a given domination number. In this paper, we solve this open problem for trees with a given domination number. Furthermore, we determine an upper bound for the Euler Sombor index of trees with a given independence number. We also characterize the corresponding extremal trees. Additionally, we propose a set of open problems for future research.

The exponential Randić index of a graph \(G\), denoted by \(ER(G)\), is defined as \(\sum\limits_{uv\in E(G)}e^{\frac{1}{d(u)d(v)}}\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). The line graph \(L(G)\) of a graph \(G\) is a graph in which each vertex represents an edge of \(G\), and two vertices are adjacent in \(L(G)\) if and only if their corresponding edges in \(G\) are incident to a common vertex. In this paper, we proved that for any tree \(T\) of order \(n\ge3\), \(ER(L(T))>\frac{n}{4}e^{\frac{1}{2}}\).

We prove the following necessary conditions for signed graphs on complete graphs to admit additive graceful labelings, thus considerably narrowing down the search for such labelings. If a signed graph on a complete graph \(K_p, ~p\not = 4\), with \(n\) negative and \(m\) positive edges, admits an additively graceful labeling, then (1) \(p\), \(p-2\) or \(p-4\) must be a perfect square, (2) If \(p\) is a perfect square then, both \(m\) and \(n\) are even, (3) If \(p-4\) is a perfect square then, both \(m\) and \(n\) are odd, (4) If \(p-2\) is a perfect square then, \(m\) and \(n\) have opposite parity and (5) The number of odd and even labeled vertices in \(S\) are \(\frac{p+\sqrt{p-i}}{2}\) and \(\frac{p-\sqrt{p-i}}{2}\) according as \(p-i\) is a perfect square, \(i=0,2\) or \(4\). We also provide an example to show that, if a signed graph \(S\) with no negative edges is additively graceful then, \(S\) as a graph need not be graceful. Interestingly, this example also settles 3 more open problems concerning gracefulness and edge consecutive gracefulness (ecg), thereby demonstrating that ecg or \(m\)-gracefulness is not a reliable measure of gracefulness.

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact enumeration of (3, 6)-fullerenes with V vertices. We also enumerate (3, 6)-fullerenes with mirror symmetry, with 3-fold rotational symmetry, and with both types of symmetry. The resulting formulas are expressed in terms of the prime factorization of V.