On squared distance matrix of complete multipartite graphs
نویسندگان
چکیده
Let $$G = K_{n_1,n_2,\cdots ,n_t}$$ be a complete t-partite graph on $$n=\sum _{i=1}^t n_i$$ vertices. The distance between vertices i and j in G, denoted by $$d_{ij}$$ is defined to the length of shortest path j. squared matrix $$\Delta (G)$$ G $$n\times n$$ with $$(i,j)^{th}$$ entry equal 0 if $$i j$$ $$d_{ij}^2$$ \ne . We define energy $$E_{\Delta }(G)$$ sum absolute values its eigenvalues. determine inertia compute More precisely, we prove that $$n_i \ge 2$$ for $$1\le \le t$$ , then $$ E_{\Delta }(G)=8(n-t)$$ h= |\{i: n_i=1\}|\ge 1$$ $$\begin{aligned} 8(n-t)+2(h-1) }(G) < 8(n-t)+2h. \end{aligned}$$ Furthermore, show fixed value n t, both spectral radius graphs are maximal split $$S_{n,t}$$ minimal Turán $$T_{n,t}$$
منابع مشابه
Eigenvalues of the resistance-distance matrix of complete multipartite graphs
Let [Formula: see text] be a simple graph. The resistance distance between [Formula: see text], denoted by [Formula: see text], is defined as the net effective resistance between nodes i and j in the corresponding electrical network constructed from G by replacing each edge of G with a resistor of 1 Ohm. The resistance-distance matrix of G, denoted by [Formula: see text], is a [Formula: see tex...
متن کاملComplete multipartite graphs are determined by their distance spectra
Article history: Received 24 October 2013 Accepted 27 January 2014 Available online 6 February 2014 Submitted by R. Brualdi MSC: 05C50 05C12
متن کاملIntegral complete multipartite graphs
A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...
متن کاملDecompositions of complete multipartite graphs
This paper answers a recent question of Dobson and Marušič by partitioning the edge set of a complete bipartite graph into two parts, both of which are edge sets of arctransitive graphs, one primitive and the other imprimitive. The first member of the infinite family is the one constructed by Dobson and Marušič.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Indian Journal of Pure and Applied Mathematics
سال: 2023
ISSN: ['0019-5588', '0975-7465', '2455-0000']
DOI: https://doi.org/10.1007/s13226-023-00386-2